Time, Heat and Work · 2010. 3. 29. · Katholieke Universiteit Leuven Faculteit Wetenschappen...

Katholieke Universiteit Leuven

Faculteit Wetenschappen

Instituut voor Theoretische Fysica

Time, Heat and Workaspects of nonequilibrium statistical mechanics

Tim Jacobs

Promotor:Prof. Dr. Christian Maes

Proefschrift ingediend tot hetbehalen van de graad van

Doctor in de Wetenschappen

Leuven 2006

Contents

Introduction 31. Thermodynamics and statistical mechanics . . . . . . . . . . . . 32. Beyond equilibrium: time enters the play . . . . . . . . . . . . . 53. Fluctuation relations for work and heat . . . . . . . . . . . . . . 74. A discussion on the results . . . . . . . . . . . . . . . . . . . . . 95. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1 Describing physical systems 151.1 Physical systems . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1.1 Scene . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.1.2 Scales of objects and levels of description . . . . . . . . 161.1.3 Minimal requirements for a description of an object . . 17

1.2 The description of classical systems . . . . . . . . . . . . . . . . 171.2.1 Microstates . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 The microscopic dynamics . . . . . . . . . . . . . . . . . 181.2.3 Distributions on phase space . . . . . . . . . . . . . . . 201.2.4 Macrostates . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3 The description of quantum systems . . . . . . . . . . . . . . . 241.3.1 Microstates . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.2 Microscopic dynamics . . . . . . . . . . . . . . . . . . . 251.3.3 Macrostates . . . . . . . . . . . . . . . . . . . . . . . . . 281.3.4 Pure states and statistical mixtures . . . . . . . . . . . 29

1.4 The description of stochastic systems . . . . . . . . . . . . . . . 321.4.1 Detailed balance . . . . . . . . . . . . . . . . . . . . . . 331.4.2 Continuous diffusion processes . . . . . . . . . . . . . . 341.4.3 Elements of stochastic calculus . . . . . . . . . . . . . . 37

2 Entropy 412.1 The trouble with entropy . . . . . . . . . . . . . . . . . . . . . 412.2 The definition of entropy . . . . . . . . . . . . . . . . . . . . . . 43

2.2.1 The historical definition of entropy . . . . . . . . . . . . 432.2.2 Thermodynamic definition of entropy . . . . . . . . . . 452.2.3 The Boltzmann entropy in classical systems . . . . . . . 472.2.4 The Shannon entropy in classical systems . . . . . . . . 522.2.5 The Gibbs entropy in classical systems . . . . . . . . . . 56

iv CONTENTS

2.2.6 The Boltzmann entropy in quantum systems . . . . . . 612.2.7 The von Neumann entropy in quantum systems . . . . . 622.2.8 The Gibbs-von Neumann entropy in quantum systems . 672.2.9 Relative entropy . . . . . . . . . . . . . . . . . . . . . . 69

2.3 Entropy as a measure of irreversibility . . . . . . . . . . . . . . 722.3.1 Different kinds of reversibility . . . . . . . . . . . . . . . 732.3.2 Time-reversal symmetry breaking and entropy . . . . . 74

2.4 The second law of thermodynamics . . . . . . . . . . . . . . . . 772.4.1 The thermodynamic arrow of time . . . . . . . . . . . . 772.4.2 Gibbs entropy and the second law revisited . . . . . . . 792.4.3 A microscopic foundation of the second law? . . . . . . 822.4.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . 84

2.5 Final words on entropy . . . . . . . . . . . . . . . . . . . . . . . 872.5.1 Entropy as a measure of disorder? . . . . . . . . . . . . 872.5.2 Entropy is contextual . . . . . . . . . . . . . . . . . . . 88

3 Quantum entropy production as a measure of irreversibility 913.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.2 Entropy production for a unitary evolution . . . . . . . . . . . 93

3.2.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.2.2 The time-reversal invariant situation . . . . . . . . . . . 943.2.3 The production of entropy . . . . . . . . . . . . . . . . . 94

3.3 Perturbations by measurement . . . . . . . . . . . . . . . . . . 953.3.1 Path-space measure . . . . . . . . . . . . . . . . . . . . 963.3.2 Time-reversal . . . . . . . . . . . . . . . . . . . . . . . . 973.3.3 Entropy production . . . . . . . . . . . . . . . . . . . . 97

3.4 Conclusions and additional remarks . . . . . . . . . . . . . . . 99

4 Aspects of quantum irreversibility and retrodiction 1014.1 The trouble with irreversibility . . . . . . . . . . . . . . . . . . 1014.2 Statistical reversibility I . . . . . . . . . . . . . . . . . . . . . . 103

4.2.1 The classical realisation . . . . . . . . . . . . . . . . . . 1034.2.2 The jump to the quantum world . . . . . . . . . . . . . 104

4.3 Quantum formalism . . . . . . . . . . . . . . . . . . . . . . . . 1054.4 Statistical reversibility II . . . . . . . . . . . . . . . . . . . . . . 1074.5 Retrodiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.5.1 The magic of retrodiction . . . . . . . . . . . . . . . . . 1104.5.2 More spectacular examples. . . . . . . . . . . . . . . . . 1114.5.3 When retrodiction seems to go wrong . . . . . . . . . . 114

4.6 Thermodynamic irreversibility . . . . . . . . . . . . . . . . . . . 1164.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5 An extension of the Kac ring model 1195.1 Relaxation to equilibrium . . . . . . . . . . . . . . . . . . . . . 1195.2 The original Kac ring model . . . . . . . . . . . . . . . . . . . . 120

5.2.1 The Stoßzahlansatz of Boltzmann . . . . . . . . . . . . 1205.2.2 A model to clarify the Stoßzahlansatz . . . . . . . . . . 122

CONTENTS v

5.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.3 An extension of the Kac ring . . . . . . . . . . . . . . . . . . . 127

5.3.1 Hilbert space and notation . . . . . . . . . . . . . . . . 1285.3.2 Dynamics and observables . . . . . . . . . . . . . . . . . 1285.3.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 1305.3.4 Initial data . . . . . . . . . . . . . . . . . . . . . . . . . 1315.3.5 Autonomy and entropy . . . . . . . . . . . . . . . . . . 132

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.5.1 Molecular chaos . . . . . . . . . . . . . . . . . . . . . . 1355.5.2 Autonomous equations and H−theorem. . . . . . . . . 1365.5.3 Relaxation to equilibrium? . . . . . . . . . . . . . . . . 1385.5.4 Higher spins . . . . . . . . . . . . . . . . . . . . . . . . . 1385.5.5 Kac ring model . . . . . . . . . . . . . . . . . . . . . . . 139

5.6 Proof of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.6.1 Equivalence of ǫ-ensembles . . . . . . . . . . . . . . . . 1405.6.2 Typical equivalence of the unitary and reduced dynamics 1415.6.3 The initial states are dispersion free. . . . . . . . . . . . 1445.6.4 The relaxation of the magnetization . . . . . . . . . . . 1455.6.5 The equilibrium magnetization . . . . . . . . . . . . . . 1465.6.6 The calculation of the Gibbs-von Neumann entropy . . 1475.6.7 Typicality explored . . . . . . . . . . . . . . . . . . . . . 148

5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6 Fluctuation relations for work and heat 1516.1 A model to study fluctuations for heat and work . . . . . . . . 151

6.1.1 The motivation to study fluctuations . . . . . . . . . . . 1516.1.2 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526.1.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.1.4 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.2 A short historical overview . . . . . . . . . . . . . . . . . . . . 1556.2.1 The first theoretical results . . . . . . . . . . . . . . . . 1556.2.2 An extension of the fluctuation theorem . . . . . . . . . 1586.2.3 Experimental verification of the fluctuation theorems . . 160

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.3.1 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.3.2 Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.4 Numerical simulations for various protocols . . . . . . . . . . . 1726.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.5.1 The harmonic potential with a general protocol . . . . . 1746.5.2 Exact identities (Crooks and Jarzynski relations) . . . . 1756.5.3 The involution s . . . . . . . . . . . . . . . . . . . . . . 1776.5.4 Exact fluctuation theorem for the work . . . . . . . . . 1786.5.5 Large deviation rate function for the work . . . . . . . . 1796.5.6 Fluctuation theorem for the heat . . . . . . . . . . . . . 179

6.6 A toy calculation for large deviations . . . . . . . . . . . . . . . 182

vi CONTENTS

6.7 The basis of a Jarzynski relation . . . . . . . . . . . . . . . . . 1856.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7 Conclusions and outlook 1897.1 Reversibility and irreversibility in the quantum formalism . . . 1897.2 Fluctuation relations and irreversibility . . . . . . . . . . . . . 1917.3 Open questions and future work . . . . . . . . . . . . . . . . . 193

A Physical constants 197

B Samenvatting 199

Bibliography 211

Index of terms 219

Introduction

1. Thermodynamics and statistical mechanics

A key moment in the technological development of the western society is with-out doubt the industrial revolution of the 19th century. This evolution wasprimarily driven by the introduction of automatic machinery into the exist-ing industry, which was possible after the fuel-saving improvements to existingsteam engines by the physicist James Watt in 1765. Shortly after this develop-ment, many experimental and theoretical studies were undertaken to improvethe efficiency of heat engines. The major leap forward from a theoretical per-spective was done by the young Sadi Carnot, with his work “Réflexions sur lapuissance motrice du feu” of 1824. Carnot presented a logical argument whichprovided an upper bound for the maximum efficiency of a heat engine. Moreimportantly, Carnot realized that these limitations to the amount of thermalenergy which can be converted into useful work, are related to the reversibilityof the engine.

Together with the pioneering work of Joule and Clausius, the study of theefficiency of engines led to the formulation of the theory of thermodynamics.This theory is built on two laws: the first law of thermodynamics expressesthat energy can be converted from heat to work and visa versa. This is usu-ally expressed as dU = dW − dQ, where dU is the total energy change of thesystem, dW is the work performed on the system, and dQ is the heat givenoff to the environment. All thermodynamic processes must verify this energybalance equation. The second law of thermodynamics puts an upper bound onthe amount of heat that can be converted into useful work. This is typicallyexpressed by introducing a new state variable called the entropy S, which isrelated to the amount of heat that is dissipated. The second law of thermody-namics then dictates that for any spontaneous process in a closed and isolatedsystem, the entropy cannot decrease. The second law is commonly expressedas dS > 0. We say that a system has reached thermodynamic equilibriumwhen the entropy has reached its maximum value, given the possible physicalconstraints.

As humans, our primary interface with nature is through the properties ofmacroscopic matter. One of the striking features of that macroscopic world

4 Introduction

is that one direction of time is singled out: people are born young, and growolder; the cup of coffee cools down but never spontaneously reheats; mixingcold and hot water gives lukewarm water, which never spontaneously splitsinto hot and cold water again. These intuitive and clear phenomena are allmanifestations of the second law of thermodynamics, which provides a frame-work for understanding the macroscopic arrow of time that we are familiarwith. Transitions which leave the entropy of the world unchanged are calledthermodynamically reversible and can occur in any direction. The equilibriumstate of the system, characterized precisely by zero entropy production, is thusa reversible state and therefore, in equilibrium there is no preferred directionof time.

The emergence of the arrow of time is very surprising. The Newtonian orSchrödinger equations of motion, which govern the microscopic world, makeno distinction between past and future: they are time-reversal invariant. Thismeans that for any solution of the microscopic equations, we can construct anew solution by replacing the time t in the equations by −t (and for the quan-tum case, we also need to do a complex conjugation). If both a time-forwardand time-backward solution are possible from a microscopic perspective, thenthe natural question to ask is why the macroscopic world prefers solutionswhich increase the entropy, which is needed to satisfy the experimentally es-tablished second law of thermodynamics.

The solution to this paradox for classical systems was already provided bythe pioneering work of Ludwig Boltzmann around the 1870s, who bridged thegap between the microscopic and macroscopic world by statistical argumentsand by recognizing the importance of the initial condititions. His microscopicdefinition of entropy, S = kB lnW , relates the thermodynamic entropy S of amacrostate with the number of microscopic states W that are compatible withthat given macroscopic state (which in turn is related to the probability ofchoosing that particular macroscopic state). The constant kB is added to getthe right units. Boltzmann’s explanation for the second law of thermodynam-ics is now a matter of comparing numbers of possible microscopic realizations.The entropy of a state where all the particles of the gas occupy one corner ofa box is very small because there are not many microscopic ways to realizethis configuration. Uniformly spreading the gas over the box increases theentropy because there are tremendously many more microscopic realizationsof this macroscopic distribution. The calculation of entropy is thus reducedto counting microscopic states and measuring phase space volumes. Togetherwith the work of Planck, Einstein and Gibbs, statistical mechanics was born.

From a reductionist perspective, statistical mechanics comes before thermo-dynamics, since it provides a microscopic description of a system. The ther-modynamic and macroscopic description can be recovered by appropriatelyreducing or coarse-graining the microscopic description. Furthermore, one hasto go through the microscopic treatment of statistical mechanics to under-

Introduction 5

stand and capture the corrections to thermodynamics in out of equilibriumsituations. Following the work of Gibbs, the microscopic distribution

ρ(x) =e−βH(x)

Z

is introduced to describe systems in thermodynamic equilibrium. Here, theinverse temperature is β = 1/kBT and H(x) is the energy of the system.

2. Beyond equilibrium: time enters the play

The theory of equilibrium statistical mechanics is well understood today inparticular thanks to the aforementioned work of Gibbs, which provides a nat-ural framework to solve, at least in principle, any equilibrium problem. Thisdoes not mean that everything has been solved to the degree of satisfaction,as is the case for many complex equilibrium problems in the field of disorderedsystems. However, for the understanding of the world around us, it is vital togo to the much larger world of nonequilibrium situations: life is an excellentexample of such a nonequilibrium system at work! No such general frameworkexists for systems outside equilibrium, which are typically systems driven bytemperature differences or by different chemical potentials. Starting with thework of Onsager in 1930, a description of such systems was provided close toequilibrium, i.e., in a linear approximation. Also this framework breaks downwhen going far from equilibrium, for example when studying explosions (ofwhich the most interesting occur in an astrophysical context), or when mod-elling living organisms. The more general problem is thus to find, if at allpossible, a general framework of statistical mechanics, which is also able tocope with nonequilibrium situations. Of course, in this thesis only aspects ofthis ambitious project are highlighted.

One first remarks that the dynamics and time-dependence of the system playan important role for macroscopic nonequilibrium systems: temperature gra-dients or differences in chemical potentials induce heat and particle currentsin the system. From the point of view of statistical mechanics, this gives riseto the question of dynamically understanding the second law, which gives thearrow of time of the macroscopic world. After all, the second law of thermody-namics is an experimental fact of the macroscopic world, and any microscopicmodelling of that world should reproduce this fact.

Again, it is the pioneering work of Ludwig Boltzmann that provides the firstanswers to this question. Boltzmann studied the mechanics of a dilute gas ofhardcore particles. The macroscopic variable of interest was the one-particlevelocity distribution f(~v, t) as a function of time. The first result of Boltzmannwas to provide an autonomous evolution equation for this macroscopic variable:

∂f

∂t+ ~v · ~∇f = ∂f

∂t

∣∣∣∣coll

,

6 Introduction

where the term on the right hand side describes the collisions between hardcoreparticles (which we will not discuss here). A second, and more surprising resultwas the so-called H-theorem. Boltzmann defined the H-functional as

H(t) =

∫

d~v f(~v, t) ln f(~v, t),

and he was able to prove that

dH

dt6 0.

Boltzmann interpreted the H function as minus the entropy, and claimed thatthis mechanical derivation gave a proof of the second law of thermodynamics.

The fact that an arrow of time arises from the time-reversal invariant micro-scopic collision laws already suggests that there are some subtleties in Boltz-mann’s derivation of the H-theorem. After a careful examination of Boltz-mann’s arguments, it turns out that the culprit is the assumption of molecularchaos, the so-called Stoßzahlansatz, which decorrelates the incoming and out-going directions of the velocities of two colliding particles in the dilute gas.However, one does not need this assumption to derive the Boltzmann equa-tion and the H-theorem: as Lanford showed in 1975 [59], it is sufficient toconsider large systems with appropriate initial conditions and by looking atthe proper timescales to recover the effective description of reality that is pro-vided by the Stoßzahlansatz and Boltzmann’s equation.

It turns out that for more general classical systems, beyond the scope of theBoltzmann equation, these three ingredients are indeed all that is needed tounderstand the emergence of the arrow of time in nonequilibrium systems.Boltzmann already realized this, and many afterwards have contributed to thebetter understanding of the increase of the Boltzmann entropy S = kB lnW .We refer to the example of an analysis by Jaynes in 1965 [64]. When startingthe system in a macroscopic state with a small W , i.e., if the initial state issufficiently special so that few microscopic states correspond to it, then thetime evolution of the system will almost certainly drive the microscopic stateof the system to a region of high phase space weight W . This is again a simplematter of counting: by comparing phase space volumes, we immediately obtainthat the system, during its dynamical excursion through the phase space, willalmost certainly end up in a macroscopic state which corresponds to a hugenumber of corresponding microstates, i.e., with a high entropy S = kB lnW .We will make this argument more precise in a later chapter, but again, itshows the importance of initial conditions, and of taking serious the fact thatwe are working with macroscopic systems. It is only for macroscopic systemswith sizes N ∼ 1023 that the differences in phase space volumes W are hugeand lead to thermodynamically different values of the entropy. The systemshould also be studied at the appropriate time scale to avoid the Poincaré re-currences of the microscopic Hamiltonian dynamics, which would violate the

Introduction 7

H-theorem. However, for the macroscopic systems under consideration, thesePoincaré recurrences occur only at time scales of the order of 1010

23

seconds,so we should not immediately worry about them.

The main conclusion is here that the second law of thermodynamics should notbe viewed as an absolute truth, but rather as a result which is typically validfor large systems under appropriate initial conditions. The above ingredientsand considerations work well in the context of classical systems. In this thesis,amongst other things, we attempt to generalize some of these arguments toclosed quantum systems by closely studying the relation between entropy andirreversibility. To do this, we will study the quantity

R = lnProbρ[ω]

Probσ[Θω],

as first introduced by Maes & Netočný [80] in the context of classical systems.This quantity R relates the probabilities to observe a trajectory ω duringa time-interval [0, τ ], starting from an initial distribution ρ = ρ0, with theprobability of observing the time-reversed trajectory Θω, starting from thedistribution σ. By choosing σ equal to the time-reversed version of ρτ , thetime-evolved measure of ρ0, the quantity R is a measure for the breaking oftime-reversal symmetry. The quest to undertake is to study the relation of Rwith entropy, H-theorems and the second law of thermodynamics.

3. Fluctuation theorems for work and heat

Another popular line of research that goes beyond equilibrium situations isthe study of so-called fluctuation theorems for dissipative quantities. The mo-tivation to study these fluctuations comes from the emerging research of nano-and biotechnology, where small devices have to operate and function under“large” disturbances from the outside world. To give an idea of the energyscales involved, we refer to molecular motors which deliver work of the orderof magnitude of a few kBT ’s while being subjected to thermal energy fluctua-tions which are typically of that same order. Intuitively, this makes clear thatwe should study the average work w and the deviations thereof, which are ofthe order of magnitude of w themselves (the so-called large deviations).

A fluctuation theorem is a statement about the symmetry properties of theprobability distribution or probability density of an observable A. Generically,it is of the form

lnProb[A = a]

Prob[A = −a] = Ca,

sometimes involving some kind of asymptotics for the system size or the ob-servation time and with C a constant. Such a relation was first observed innumerical experiments by Cohen, Evans and Morris [43], which led to the rig-orous formulation of a fluctuation theorem for the phase space contraction rate

8 Introduction

by Gallavotti and Cohen [48, 49]. Within the context of chaotic dynamical sys-tems, the role of the observableA is played by the phase space contraction ratioṠ, which in turn is viewed in these references as the entropy production. Thisbrings us back to the study of irreversibility. However, fluctuation relationsfor the entropy production give more information about how the typicality ofthe irreversibility, i.e., the increase of entropy, is attained. It does not onlytell us that the probability Prob[Ṡ = λτ ] for a positive entropy productionṠ > 0 is much more likely than decreases in entropy, Ṡ < 0, but also thatthe typical behaviour as discussed by Boltzmann is attained in an exponentialfashion, in the Gallavotti-Cohen case asymptotically in the observation time τ .

Recently, van Zon and Cohen studied in a series of papers [121, 122, 123] aparticle immersed in a fluid, and dragged by a harmonic potential at a constantvelocity. Due to the collisions with the molecules in the fluid, the particle’sposition, and hence also the derived quantities such as the work delivered onthe particle and the heat dissipated by the particle to the environment, is arandom variable. Van Zon and Cohen modelled the particle with Langevin dy-namics and concluded after an explicit calculation that the work Wτ deliveredon the particle by an external agent satisfies, up to a multiplicative constant,the fluctuation theorem

lnProb[Wτ = wτ ]

Prob[Wτ = −wτ ]= wτ, w > 0.

This relation expresses that the probability for the unexpected event that theparticle does work on the external agent, i.e., Wτ < 0, is exponentially dampedin the observation time τ and thus becomes increasingly more unlikely.

The first law of thermodynamics asserts that the heat Qτ differs from thework Wτ by the energy difference ∆U . One would expect that asymptoticallyqτ = Qτ ≈ Wτ = wτ since ∆U/τ vanishes for large τ . Hence, it seemstempting to extrapolate the fluctuation theorem for the work to an asymptoticfluctuation theorem for the heat:

limτ→∞

1

τln

Prob[Qτ = qτ ]

Prob[Qτ = −qτ ]= q.

Interestingly, this is not what happens: there is a nonzero probability thatthe particle stores the work Wτ done on it in the form of energy ∆U , thuspreventing ∆U/τ of going to zero as τ → ∞. This rare event gives rise to acorrection of the fluctuation theorem. Again by explicit computations, van Zonand Cohen obtained that the heat satisfies the following extended fluctuationtheorem:

limτ→∞

1

τln

Prob[Qτ = qτ ]

Prob[Qτ = −qτ ]=

q, for q ∈ [0, q[,q − (q − 1)2/4, for q ∈ [q, 3q[,2, for q > 3q,

Introduction 9

where q is the mean heat dissipation per unit time. Here, we see that smallfluctuations behave as one would expect in a fluctuation theorem. Howeverthe large fluctuations, which are more than 3q, are all equally damped in theobservation time τ , i.e., very large fluctuations in the heat are more likely thanthose in the work.

The case of the dragged harmonic potential is special in the sense that every-thing can be computed explicitly; this is possible because the work turns outto have a Gaussian distribution. Immediately, the question arises whether thecorrection to the fluctuation theorem discussed above is a peculiarity of theharmonic case, or whether it is more generally valid. Furthermore, it seemsessential to understand why and under what circumstances such a correctionfor the heat can arise if one seriously wants to study heat dissipation, irre-versibility and entropy production. That is exactly what we have done in partof the present thesis.

4. A discussion on the results

Reversibility and irreversibility in the quantum formalism

The first result of this work concerns the relation between entropy and thebreaking of time-reversal invariance in closed quantum systems. First, wehave to define what we mean by a quantum trajectory of macroscopic states.The microscopic evolution, as determined by the Schrödinger equation, givesrise to an evolution at the level of macroscopic variables (the reduced or coarse-grained description). For example, we can study as a function of time the valueof the total magnetization of a magnetic system as the macroscopic variable,while it changes under the microscopic evolution determined by a Hamiltonian.By performing measurements on the system at given times, we can write a se-quence of macroscopic states that the system has visited.

Using this notion of macroscopic trajectories, we can understand the relationbetween observing a particular macroscopic trajectory ω during a time-interval[0, τ ] and its time-reversed variant Θω. It turns out that also in a quantummechanical description, the breaking of time-reversal symmetry can be relatedto the quantum Boltzmann entropy and Gibbs-von Neumann entropy. Roughlyspeaking, one has

lnProb[ω]

Prob[Θω]= SB(ωτ ) − SB(ω0),

where SB(α) is the quantum Boltzmann entropy. An immediate consequenceis the positivity of the entropy production in closed quantum systems, un-dergoing an evolution which is interrupted by measurements to determine themacroscopic trajectory. From this, we recover statistical reversibility, a condi-tion similar to that of detailed balance, which in turn is known to express the

10 Introduction

reversibility in standard stochastic models.

In standard textbook quantum mechanics, the underlying microscopic quan-tum dynamics is a combination of the reversible unitary time evolution asdescribed by the Schrödinger equation, and the fundamentally irreversible col-lapse of the wave function. As such, there is no reversibility at the micro-scopic level since it is impossible to “unmeasure” or “decollapse” the system.However, as the result above shows, all this does not prevent the recovery ofstatistical reversibility.

On the other hand, the microscopic reversibility of the unitary evolution ofquantum mechanics does not contradict the emergence of thermodynamic ir-reversibility. In a toy model, we study the magnetization of N distinguishablespin-1/2 particles as a function of time, undergoing a standard unitary evo-lution. We are able to construct an irreversible, effective and autonomousdynamics at the level of the macroscopic variables, inspired by (but not using)Boltzmann’s assumption of molecular chaos or Stoßzahlansatz. We prove thatfor almost all conceivable realizations of our model, and for a large class ofinitial conditions, the unitary evolution converges to the effective dynamics asthe system size grows. This leads to the conclusion that the reasons for ther-modynamic irreversibility are unchanged when going from the classical to thequantum world: it is again an interplay between the fact that we are workingwith large systems, using appropriate initial conditions and by looking at theproper timescales, in this case to avoid the quasi-periodicity of finite (but verylarge) quantum systems.

Fluctuation theorems for heat and work

In the mathematical context of a driven Langevin dynamics, we study a par-ticle under influence of a potential Ut that is changed according to a givenprotocol γt. The observables of interest are the work performed on the particleby the external agent that is modifying the potential, and the heat dissipatedto the environment by the particle.

The conceptual framework is again that of looking at the breaking of time-reversal symmetry. A first result is the recovery of the so-called Crooks re-lation, which identifies the dissipation function and source of time-reversalsymmetry breaking as the dissipative work W dis = W −∆F . This dissipativework is the total work W performed on the system, minus the reversible work∆F , and as such it gives the energy lost through dissipation. The Crooksrelation is valid for all potentials Ut and all protocols γt.

It is known that there exist experimental set-ups in which no fluctuation the-orem for the work is found. Therefore, an additional condition is neededto derive that fluctuation theorem from the Crooks relation, and when thiscondition is not verified, the fluctuation theorem should not hold. It turns

Introduction 11

out that this additional condition is most conveniently expressed in terms ofspatio-temporal symmetries of the potential and protocol: when the protocolis antisymmetric under time-reversal and the potential is symmetric under spa-tial reflections, or in case that the protocol is symmetric under time-reversalbut then for any potential, we recover the fluctuation theorem for the work.Furthermore, we also find that the harmonic case of van Zon and Cohen isindeed special: the work always satisfies a fluctuation symmetry, regardlessof spatio-temporal symmetries of the potential and protocol. Using numericalsimulations, we are able to verify that the conditions for the existence of afluctuation theorem are in fact optimal: from the moment they are not satis-fied, deviations from the fluctuation theorem can occur.

For the heat Q = W − ∆U , we must understand under which circumstancesthe large deviations for ∆U give contributions of the order of the observationtime τ to find back deviations from the fluctuation theorem that is valid forthe work. We present a general argument based on the large deviation ratefunction of the work and the energy distribution ∆U . For general potentials,we recover the three regimes that were discussed by van Zon and Cohen. Inour case, we can see clearly how the large deviations for the work are connectedto the saturation of the fluctuation relation for large values of the heat perunit time q.

However, the main conclusion in the study of fluctuations is that again theframework of the time-reversal breaking is successful and unifies several re-cently proven results such as the Jarzynski equality, Crooks relation and fluc-tuation theorems in a broader picture.

5. Outline

This section briefly summarizes the content of each chapter and provides thegrand master plan behind this thesis.

The first chapter contains a brief overview of some of the mathematical toolsthat are used in this work. We quickly review some aspects of classical Hamil-tonian mechanics and quantum mechanics. The emphasis there is on the dis-tinction between microscopic and macroscopic descriptions. Also some aspectsof continuous diffusion processes for stochastic descriptions are discussed. Itis not our goal to be very rigorous in the short overview that we present; theemphasis is on the conceptual framework, and when necessary, we will givereferences to other works for more details.

Chapter two presents a discussion of the concept of entropy, which will playa central role throughout this work. First, we go into detail on the defini-tions of entropy that exist in the literature: the relations between the classicalClausius entropy, thermodynamic entropy, Boltzmann entropy, Shannon andGibbs entropy will be discussed. We also give the quantum Boltzmann en-

12 Introduction

tropy, the von Neumann and von Neumann-Gibbs entropy, and summarize allentropies in the definition of relative entropy on the level of classical measuresand density matrices. The chapter concludes with some words on the knownrelation between entropy and irreversibility in classical systems, and how weshould understand the second law of thermodynamics. Again, it is not theintention to give a complete overview in this chapter, because much more canand should be said about the subtleties regarding the definition of entropy.On the other hand, the central role of entropy in the study of irreversibilityforces us to review some results from the literature.

The third chapter proposes a relation between entropy and time-reversal sym-metry breaking in quantum systems. By looking at trajectories of macroscopicstates, constructed by measuring at given times the macrostate of the system,we hope to find that the quantum Boltzmann entropy, which in some waymeasures the “quantum phase space volume”, measures the irreversible be-haviour of the quantum system. The concept of detailed balance, as knownfrom stochastic models, will play an essential role in understanding the emer-gence of irreversibility in quantum systems.

After that, we turn in chapter four to the issues of the collapse of the wave func-tion and retrodiction in quantum mechanics. The fundamental irreversibilityof the collapse of the wave function seems to prevent microscopic reversibilityof entering the quantum description. We will argue why the inability to undomeasurements does not necessarily contradict the statistical reversibility at thelevel of trajectories, introduced in chapter three. To conclude that chapter, wealso present the difficulties of retrodiction in quantum mechanics, and how thisgives rise to seemingly paradoxical statements about simultaneous knowledgeof noncommuting quantum observables.

Chapter five presents a simple quantum toy model which illustrates the con-cepts of the preceding two chapters. This model is a generalization of the Kacring model, introduced in 1956 [66] to understand the reversibility paradoxand the status of the Boltzmann H-theorem. The classical Kac model illus-trates in detail that there is no contradiction between macroscopic irreversiblerelaxation to equilibrium and the dynamical reversibility of the microscopicequations of motion. We lift this model to the quantum level and retrievesimilar conclusions regarding irreversibility in quantum systems.

In chapter six, we turn back to classical systems and look at fluctuation re-lations for the work and heat in small systems. The goal is here to studyLangevin systems which are more general than the dragged harmonic poten-tial of van Zon and Cohen. We provide a statement about the conditions forthe existence of a fluctuation theorem for the work, and we can then under-stand when to expect corrections to a fluctuation theorem for the heat. Thischapter also gives a review of some of the experimental tests of the knownfluctuation theorems.

Introduction 13

Finally, in chapter seven, we summarize the results by giving the conclusionsof this work and we end by discussing some possible future extensions of thisthesis.

14 Introduction

Chapter 1

Describing physicalsystems

In the description of the reversible or the irreversible nature of physical phe-nomena, the scale will turn out to play a crucial role. In this chapter, wediscuss some different scales of reality, and we explain how to attach a clas-sical or a quantum mechanical framework to the corresponding reality. Wecomment in a few lines on the history of decriptions of nature, to move alongquickly to some of the mathematical frameworks that are known and used today.

1.1 Physical systems

1.1.1 Scene

Physics is about describing and unifying physical phenomena. That refers tothe world around us, as composed of so-called physical objects. A remarkabledefinition of a physical object is read in Webster’s dictionary, which defines itas...

... an object having material existence: perceptible especiallythrough the senses and subject to the laws of nature.

The reference to “material existence” is very much alive in contemporaryand popular discussions, but goes in fact back to Aristotle, who lived around350 B.C. That matter is composed of more elementary and universal build-ing blocks such as atoms was first conceived by the Greek atomists such asDemocritus, around 400 B.C. At any rate, statistical physics talks about theemergence of the world around us out of those rawest material building blocks,and how it appears in mesoscopic or macroscopic conditions. Clearly, that rep-resents an enormous bridging of scales. Roughly speaking, we are interestedin the global behaviour of about 1023 particles as subject to the (microscopic)laws of nature, expressing that nature can be described by rational and logi-

16 CHAPTER 1 - Describing physical systems

cally consistent statements which predict how one state of the object evolvesinto a new state.

1.1.2 Scales of objects and levels of description

Before going into the question on how to characterize a physical system, thelevel of detail of the description must be fixed.

The number of constituents of a physical object or system determines an im-portant scale, which in turn is related to the more visible scales such as time-,length- or energyscales. We restrict ourselves to two scales of objects in thisthesis:

• microscopic objectswhen less than approximately 104 – 106 elementary constituents are in-volved in the composition of the object.

• macroscopic objectswhen at least of the order of 1023 elementary constituents are involvedin the composition of the object.

Most objects that we can handle, feel and touch with our hands are typicallycomposed of 1023 atoms or molecules. The mesoscopic scale of approximately106 to 1023 constituents will not be discussed here.

Though multiple levels of description exist, we shall restrict ourselves to twolevels in this thesis:

• microscopic or complete descriptionwhere complete knowledge of all the 1023 elementary constituents of thephysical object is required before the description is considered to becomplete.

• macroscopic or reduced descriptionwhere only global properties concerning all or some of the elementaryconstituents of the physical object are required for the description to beconsidered complete.

From an everyday point of view, the macroscopic description is the most com-mon way of expressing the state of a physical object: it is sufficient to give thelocation, volume and temperature of a glass of water to have a good idea whatthe physical object under consideration looks and behaves like (water vapour,liquid water, ice).

Strictly speaking, there is no one-to-one mapping between the level of descrip-tion and the scale of the object under consideration. Obviously, microscopicdescriptions work well for microscopic objects, and macroscopic descriptionsfor macroscopic objects. It is theoretically possible to give a microscopic (com-plete) description of a macroscopic object or to try to attach a macroscopic

The description of classical systems 17

description to a microscopic object. From a practical point of view, this isuncommon: a microscopic description of a macroscopic object requires theknowledge of about 1023 constituents and is currently beyond human andcomputational capabilities.1 Attempting to connect macroscopic descriptionsto microscopic objects typically leads to interpretational problems of conceptslike temperature, density or pressure.

1.1.3 Minimal requirements for a description of an object

For both microscopic and macroscopic descriptions, there are usually threeingredients that are used, for example, to make predictions about the physicalobject under consideration:

1. The initial state of the system which is the starting point for thedescription. Specifying the initial state also requires specification of thelevel of description (macroscopic or microscopic) and of the phase space.

2. The dynamicswhich is required to specify how the initial state will evolve into anotherstate of the system, and to determine which states are left invariant bythe dynamics.

3. The physical properties of interestor observables that we want to concentrate on.

The possible choices for these three ingredients depend on whether a classicalor a quantum mechanical description is preferred. In the following sections,we will discuss both scenarios.

1.2 The description of classical systems

1.2.1 Microstates

The microscopic state x, or in short “microstate”, of a classical system iscompletely determined by specifying all the properties of each of the N con-stituents (which we will refer to as particles). We give two commonly usedexamples:

• To describe a mechanical system, the position ~qj ∈ R3 and momen-tum ~pj ∈ R3 of all particles j ∈ {1, . . . , N} should be specified. Themicrostate x describing the entire system is then given by

x = (~q1, ~q2, . . . , ~qN , ~p1, ~p2, . . . , ~pN ) ∈ R6N .We will usually group all positions and momenta together for a lighternotation and write x = (q, p).

1Nobody has succeeded up to today to give the full microscopic description of thatparticular glass of water, and the author remains confident that nobody will for quite sometime.


• To describe a magnetic system on a lattice Λ ⊂ Zd, the (classical) spin ~sjor the intrinsic magnetic moment ~µj = µB~sj (where the Bohr-magnetonµB is the proportionality constant) of each particle should be specified.The microstate x is then given by

x = (~s1, ~s2, . . . , ~sj) ∈ SN ,

where S is the set of all possible (classical) spin values. For example, withIsing spins in one dimension, the possible spin values are S = {−1,+1}.Remark that magnetism is a quantum phenomenon, but an excellentclassical caricature can be obtained by the Ising “classical” spins. Howto describe a full quantum spin is discussed in section 1.3.1.

In general, it is sufficient to provide the phase space Γ, which is the set of allpossible microstates x ∈ Γ and possibly subject to external constraints (suchas particles in a box) and to internal conditions (such as fixing the energy).The specific interpretation of the microstate x is irrelevant for the followingsections.

1.2.2 The microscopic dynamics

The phase space Γ provides us with one of the three ingredients to describe asystem: states. To make things interesting, an evolution between these statesmust also be defined.

Hamiltonian dynamics

The Hamiltonian H(x, t) is a function from the phase space Γ into the realnumbers R which, under standard conditions, represents the energy of thesystem under consideration. For mechanical systems, a typical Hamiltonian isof the form

H =

N∑

j=1

~pj2

2m+∑

i


level of the phase space, we can see the set of Hamilton equations as a flow ftwhich evolves the initial condition through

ft : Γ → Γ :x0 7→ xt.

(1.2)

The Hamiltonian flow is invertible: the inverse f−1t exists, and one can go backfrom a point xt to the corresponding initial condition x0 by applying f

−1t to

xt, see figure 1.1.

x 0x t

ft

x 0x t

ft-1

Figure 1.1: The inverse f−1t of the Hamiltonian flow ft allows to go back to the initialstate x0 when applied to the final state xt.

Consider a classical mechanical system going through a sequence of positionsand momenta x0 = (q0, p0), . . . , xt = (qt, pt). That evolution solves Hamil-ton’s equations of motion (1.1) for given forces, e.g., gravity. Upon playingthe movie backwards, i.e., time reversed, we see the system evolving from thepositions qt to q0, but now with reversed momenta −pt to −p0.

In general, the time-reversed sequence (qt,−pt), . . . , (q0,−p0) can or cannot bea solution of the same equation of motion. For say free fall, the time-reversedsequence certainly solves the same Newton’s law of gravity; with friction or forthe damped oscillator, that time-reversal symmetry is not present (but thenwe are no longer in the case of a Hamiltonian dynamics).

The Hamiltonian dynamics is always invariant2 under time-reversal, i.e., thereplacement of the time t by −t in the equations (1.1). Time-reversal changesthe sign of the momentum

~pj =d~qjdt,

which represents the fact that we are playing the movie backwards. We for-mally define a time-reversal operation π : Γ → Γ through3

π : x = (q, p) 7→ πx = (q,−p). (1.3)

The time-reversibility of the Hamiltonian flow can then be formulated as

πftπ = f−1t .

2For systems under the influence of magnetic fields, also the direction of the magneticfield should be inverted.

3For a Hamiltonian system in a magnetic field, the time-reversal operation becomesπ(q, p,A) = (q,−p,−A).


In words: the inverse f−1t of the Hamiltonian flow can be achieved by changingthe sign of the momenta in xt, and then applying the time-forward evolutionto πxt; to restore the proper initial state x0, the momenta must be reversedonce again (see also figure 1.2).

x 0x t

ft-1

πx t

ft

ft πx t

Figure 1.2: Up to a change of momenta, the inverse evolution f−1t can be attained byapplying the regular flow ft to πxt.

General dynamical systems

In general, e.g., for non-Hamiltonian or nonmechanical systems, the time evo-lution of the microscopic variables is of the form

dx

dt= F (x) (1.4)

and defines a flow ft on the phase space Γ; thus, we can still write xt = ftx0as in the Hamiltonian case. However, the inverse of this general flow ft doesnot need to exist.

The notion of time-reversal invariance can be recovered for invertible flows.

Definition 1.2.1An invertible dynamical flow ft is called time-reversal invariant if and onlyif there exists an involution π, with π2 = 1, such that

πftπ = f−1t . (1.5)

The involution π is referred to as the kinematic time-reversal or microscopictime-reversal.

If the phase space Γ has further structure, such as carrying a natural metric,then the involution π is supposed to preserve that additional structure.

1.2.3 Distributions on phase space

In practical situations, it is often not well known what precise microscopicstate the system is in: macroscopic systems require the knowledge of about1023 positions and momenta! In those situations, it is often more convenientto work with a probability distribution on the microscopic phase space Γ.


Let ρ(x) be a probability density on the phase space, in the sense that

dρ(x) = ρ(x)dx

gives the probability for the system to be found in a microstate in the in-finitesimal phase space volume dx. The reference measure dx can be theLebesgue/Liouville volume element or is derived from another natural metric.We consider here the classical statistical mechanics with dx = d~q1 . . . d~pN . Fora given initial distribution ρ0(x) on the phase space, the Hamilton equations(1.1) can be lifted to an evolution equation for the measure. Using the Liouvilletheorem

dρtdt

= 0,

we obtain the Liouville equation

∂ρt∂t

+

N∑

j=1

∂ρt∂qj

q̇j +∂ρt∂pj

ṗj = 0,

which is usually4 written in terms of the Poisson bracket {·, ·} as

∂ρt∂t

= −{ρt, H}, {A,B} ≡N∑

j=1

∂A

∂qj

∂B

∂pj− ∂B∂qj

∂A

∂pj. (1.6)

Because the Liouville equation implements the Hamiltonian flow (1.2), we have

ρt(ftx) = ρ0(x). (1.7)

This relation expresses that keeping the initial distribution and evolving thephase points, is equivalent to evolving the distribution in time and keeping thephase points fixed.5

An important result in classical mechanics is that the Liouville volume

| · | : A 7→ |A| =∫

A

dx, ∀A ⊂ Γ, (1.8)

with dx the Lebesgue measure (or in this context sometimes also called theLiouville measure), is preserved under the Hamiltonian flow (1.2)

|ftA| = |A|. (1.9)

4Another common way of writing the Liouville equation is

∂ρt

∂t= −Lρt,

where L = {·,H} is the Liouville operator or Liouvillian.5In the quantum case, we would refer to this as the combination of the so-called

Schrödinger and Heisenberg pictures.


This is an expression of the fact that the Lebesgue measure dx, with proba-bility density ρL(x) = 1, is invariant under the Liouville evolution (1.6).

The Liouville measure and Liouville volume are important for Hamiltoniandynamics precisely because they give an unbiased and invariant way of com-paring the weight of sets in the phase space Γ. For a general dynamics (1.4),we will usually assume the existence of an invariant measure dρ, and definethe phase space volume ρ(A) of A ⊂ Γ as

ρ(·) : A 7→ ρ(A) =∫

A

dρ. (1.10)

1.2.4 Macrostates

Consider a classical, closed and isolated system of N particles. A completedescription consists in specifying its microstate, which is a point x ∈ Γ in itsphase space.

We wish to construct a reduced description which perhaps does not take intoaccount all the details of the microscopic description. One obvious way to dothat is to erase “information” from the level of description by specifying a mapx ∈ Γ → f(x) ∈ F where F is some metric space. The map f is typically many-to-one and F is often some subspace of Rn. A somewhat simpler picture ofthat corresponds to dividing the phase space Γ into regions M ⊂ Γ, where, forexample, on each subset M the observables f1(x) . . . fm(x) take approximatelythe same values6 for all x ∈ M . We write in such a case M = (f1, . . . , fm)where f1 = f1(x) for all x ∈M . By doing this, each microstate x determines amacrostate M(x) corresponding to that much coarser division of phase space:

M : x→M(x).

Evidently, in the simplest case and to avoid ambiguities in the reduced de-scription, all the subsets Mα ⊂ Γ with (α = 1, . . . , n), should form a finitepartition of Γ, i.e.,

⋃

α

Mα = Γ, Mα ∩Mα′ = 0 for α 6= α′. (1.11)

We group the macrostates Mα in the reduced phase space Γ̂ = {M1, . . . ,Mn}.Note that this partitioning could still depend on the number of particles N inthe system.

We emphasize that this reduced description involves macroscopic variables(in contrast with the partitions as used, e.g., in constructions of dynamical

6For observables f which take values in a continuous interval [a, b] ⊆ R, we should dividethe total interval [a, b] into smaller pieces [aj , bj ] ⊂ [a, b] and assign a macrostate M(x) = Mjcorresponding to an interval [aj , bj ] to x ∈ Γ if f(x) ∈ [aj , bj ].


entropies), such as position or velocity profiles and we denote by ρ(M) thecorresponding phase space volume (1.8) or (1.10).

Since π is an involution, any phase space point x ∈ Γ can be uniquely writtenas the kinematic time-reversal of another phase point y ∈ Γ: x = πy. It is easyto find this y, since y = πx by the involution property π2 = 1. This one-to-onemapping between a phase point x ∈ Γ and πx leads to the invariance of theLiouville volume under time-reversal for macroscopic states:

|πMα| = |Mα|. (1.12)

The microscopic flow ft : x0 7→ xt, which can be Hamiltonian as in equation(1.2), implies a dynamics at the level of the reduced phase space Γ̂, wherenow M0 ≡ M(x0) 7→ M(xt). A priori, there is no reason to exclude thatthe image ft(M0) of a macrostate M0 ⊂ Γ overlaps with several macrostates,i.e., that there exist multiple Mα ∈ Γ̂ for which ft(M0)∪Mα 6= ∅, see figure 1.3.

Γ

M0

ft(M0)

ft

Figure 1.3: The phase space Γ is divided into smaller cells M ∈ Γ̂ that form the macroscopicstates. Usually, this division is inspired by a set of macroscopic observables taking constantvalues on each cell M . The microscopic flow ft (which does not need to be Hamiltonian)evolves one macrostate M0 into its image ft(M0), which a priori can overlap with several

macrostates Mα ∈ Γ̂.

However, it can happen that the dynamics on the level of the macrostates isautonomous, which means that from (only) knowing the macrostate M0 atsome given time, we also know the macrostate Mt at a later time. Typically,the separation between the microscopic and macroscopic world manifests itselfin the form of hydrodynamic or thermodynamic limit, where the number ofparticles N → ∞. It is only in this limit that we expect that the macroscopicdegrees of freedom detach from the microscopic world and that autonomy is


established. In that case, the image ft(M0) of the macrostate M0 is obviouslyconcentrated in the macrostate Mt, and hence, if only formally,

ft(M0) ⊂Mt ⇒ ρ(ft(M0)

)6 ρ(Mt),

where again, ρ is the invariant measure for the flow ft.

Even though autonomy seems like a very serious restriction on the level of themicroscopic dynamics, it is what we expect to find in the macroscopic world, atleast for a “good choice” of macroscopic observables. For example, Newton’sequations of motion for macroscopic objects subject to friction, the Navier-Stokes equations for fluid dynamics or the Boltzmann equation for dilute gases,are all for a large part independent of the details of the microscopic world. Yet,they successfully give an autonomous, macroscopic description of reality.

1.3 The description of quantum systems

1.3.1 Microstates

In quantum mechanics, the full description of a system is no longer given byall the positions and momenta of all particles, but by the wavefunction |ψ〉from a Hilbert space H, i.e., a complete, normed vector space with the norminduced by a scalar product. It has been argued that this description does notalways provide the full picture, but we will not consider that question here.We will usually assume that H is finite dimensional.

Single particle Hilbert space

The precise form of the Hilbert space depends on the degrees of freedom ofthe quantum system that are taken into account. The spatial degrees of free-dom are described by a normalized, square integrable wave function ψ(q, t) inL2(R3, dq). Following Born’s statistical interpretation of the wave function,we interpret

|ψ(q, t)|2dqas the probability for finding the particle in an infinitesimal volume [q, q+ dq]at the time t. Obviously,

∫R3 |ψ(q, t)|2dq = 1is now the normalization of the probability distribution.

Since Pauli’s theoretical work and as, e.g., manifested in the Stern-Gerlach ex-periment, a quantum description also sometimes involves an intrinsic quantityknown as spin. For our purposes, it is sufficient to say that the quantum spinis described by a normalized vector ~s ∈ C2j+1. We will be happy to deal only

The description of quantum systems 25

with the most relevant case of j = 1/2, so the Hilbert space is C2.The total wave function for the system is then given by

|ψ〉 = ψ(q, t) ⊗ ~s ∈ H, H = L2(R3, dq) ⊗C2j+1.To determine the expectation values of position and momentum, the corre-sponding operators must be defined. In the position representation, the mo-mentum operator is given by

pm = −i~∂

∂qm, m ∈ {x, y, z}. (1.13)

The operators that give the three components of a spin 1/2 particle are, up toa constant ~/2, given by the Pauli matrices:

σx =

(0 11 0

)

, σy =

(0 −ii 0

)

, σz =

(1 00 −1

)

.

In general, we write A for the algebra of observables working on the singleparticle Hilbert space H.

Multiple particles

For quantum systems composed of N distinguishable particles or components,the Hilbert space HN that contains the wave functions |ψ〉 that describe theentire system, is built up from the single particle Hilbert space H = Hj forthe jth particle:

HN =N⊗

j=1

Hj .

We write AN for the algebra of observables working on HN .

1.3.2 Microscopic dynamics

The Schrödinger equation of quantum mechanics

The Hamiltonian H is an operator on the Hilbert space HN and representsin some way the energy of the system under consideration. In the standardset-up and Schrödinger representation, a typical Hamiltonian is of the form

H =N∑

j=1

− ~2

2m

∂2

∂~qj 2+∑

i


with the formal solution

|ψt〉 = Ut |ψ0〉 , Ut = exp(

− itH~

)

. (1.15)

The unitary evolution Ut implements the evolution flow on the algebra ofobservables A as

At = U†tAUt, (1.16)

where † represents the operation of Hermitian conjugation and A ∈ A.

Time-reversal invariance of the Schrödinger evolution

One of the most visible formal differences between the Schrödinger equationand Newton’s law is that the Schrödinger equation (1.14) is first order intime while Newton’s F = ma is second order.7 Clearly then, the fact that|ψt〉 = ψ(q, t) is a solution of the Schrödinger equation (1.14) does not implythat |ψ−t〉 = ψ(q,−t) is also a solution, as was the case for Newton’s or Hamil-ton’s equations (1.1). In that very strict sense, Schrödinger’s equation is nottime-reversal invariant.

We hasten to give the standard response, that one should also complex conju-gate: |ψt〉 is a solution if and only if |ψ⋆t 〉 is a solution. One argument comesfrom the representation of the momentum (1.13), where the complex conjuga-tion switches the sign of the momentum. One could reply to that by notingthat there is no a priori reason that the momentum should change sign undertime-reversal: after all, in equation (1.13), there is only a spatial (and no time-)derivative. Furthermore, it is not clear in general how to realize experimen-tally a complex conjugation on the wave function of a system. Nevertheless,the more fruitful response is to complement time-reversal with a certain oper-ation on wave functions much in the spirit of the kinematic reversibility (1.5)as we will now explain.

One of the advantages of the abstraction around the definition of kinematicreversibility (1.5) is that it also applies to the free evolution of the quan-tum formalism, i.e., the evolution on wave functions as given by the stan-dard Schrödinger equation (1.14). Following the proposal of Wigner [128],the recipe for time-reversal is to apply complex conjugation. More generally,the transformation π of above is now an antilinear involution on the Hilbertspace, π2 = 1. We get time-reversal symmetry when that π commutes withthe quantum Hamiltonian H . Since the Schrödinger evolution is given by theunitary (1.15), equation (1.5) can now be written as

πUtπ = U†t = U

−1t , (1.17)

7One could argue that Schrödinger’s equation consists of two first order equations (sinceψ is complex), very much analogous to Hamilton’s equations of classical mechanics. Thisdoes not diminish the fact that there is still an essential difference.


where through the complex conjugation of the momentum (1.13), we still haveπ(q, p) = (q,−p), albeit through a different mechanism than in classical me-chanics.

We conclude that not only the (classical) Hamiltonian equations, but also theSchrödinger equation8 are effectively invariant under dynamical time-reversal:for the free quantum flow, future and past are mere conventions and can bedescribed by the same laws. Once the measurement procedure of collapsingthe wave function enters, we no longer have a free quantum evolution. Then,things get more complicated as we will discuss in chapter 4.

The von Neumann measurement postulate

To give the full time evolution of a quantum system, the Schrödinger equation(1.14) is usually supplemented with the collapse of the wave function. VonNeumann, known for his rigorous mathematical approach to the foundationsof quantum mechanics, argues that in order for quantum mechanics to be afundamental theory, it must also be able to describe the act of measurement[125, p418]:

First, it is inherently entirely correct that the measurement orthe related process of the subjective perception is a new entityrelative to the physical environment and is not reducible to thelatter. Indeed, subjective perception leads us into the intellectualinner life of the individual, which is extra-observational by itsvery nature[...] Nevertheless, it is a fundamental requirement ofthe scientific viewpoint – the so-called principle of the psycho-physical parallelism – that it must be possible so to describe theextra-physical process of the subjective perception as if it were inreality in the physical world – i.e., to assign to its parts equiva-lent physical processes in the objective environment, in ordinaryspace.

– John von Neumann

The measurement in quantum mechanics collapses the wave function to theeigenstate of the observable being measured, corresponding to the eigenvaluethat was measured. Write the spectral decomposition of an observable A ∈ Aas

A =

n∑

j=1

ajPaj ,

where aj ∈ R are the eigenvalues of A and Paj = P 2aj is the projector onthe eigenspace corresponding to aj. Von Neumann postulates that when themeasurement of the observable A on the wave function |ψ〉 yields the eigenvalue

8Or, for that matter, Dirac’s equation. We do not wish to speak about time-symmetrybreaking due to the weak interaction.


aj as an outcome, then the after-measurement wave function |ψ′〉 is given by,see [24]:

|ψ′〉 = Paj |ψ〉√〈ψ|Paj |ψ〉

. (1.18)

The probability to measure the eigenvalue aj as the outcome is postulated tobe

Prob(A = aj) = 〈ψ|Paj |ψ〉 . (1.19)This shows that the measurement is the point where stochasticity enters intothe quantum dynamics: at the point of measurement, the system randomlychooses one of the eigenstates of the measured observable A to collapse to, ofcourse verifying the probability distribution Prob(A = aj).

1.3.3 Macrostates

As in the classical case, we would like to split the Hilbert space into smallerparts Mα ⊂ H corresponding somehow to the joint eigenspaces of a set ofmacroscopic observables (A1, . . . , Am). The variables defining the macrostateare not different from that in the classical situation. For example, they specify(again to some appropriate accuracy) the particle number and the momentumprofiles by associating numbers to a family of macroscopically small but mi-croscopically large regions of the volume occupied by the system.

However, this is not trivial: in the quantum case, the observables which areused for the division of the Hilbert space might not commute. Therefore, theconcept of joint eigenspace is rather suspect. One says that it is not possible tomeasure noncommuting observables simultaneously. This makes it hard froman experimental viewpoint to determine if a wave function |ψ〉 should belongto a certain Mα ⊂ H. In this context, von Neumann [125, p.402] remarkedthat:

Now it is a fundamental fact with macroscopic measurementsthat everything which is measurable at all, is also simultaneouslymeasurable, i.e., that all questions which can be answered sepa-rately can also be answered simultaneously.

– John von Neumann

From this we conclude that the problem should disappear in the thermody-namic limit. We refer the interested reader to reference [32] for an extensivediscussion.

This problem vanishes, or is expected to vanish, for very large systems, forwhich N → ∞. For intensive observables of the form

AN =1

N

N∑

j=1

Aj ,


where Aj works at a small number of sites, we find

[AN , BN ] = O(

1

N

)

→ 0.

Therefore, for large systems, we can safely assume that the macrostate isgiven in terms of the values of a set of macroscopic observables representedby commuting operators. We refer to reference [32] for more details on theconstruction of the microcanonical quantum ensemble.

Then, the macroscopic partition in the classical case (1.11) is replaced by theorthogonal decomposition of the N -particle Hilbert space H:

H =⊕

α

Hα (1.20)

into linear subspaces. These linear subspaces can be found common to all thecommuting operators that have been chosen as the macroscopic variables forthe partitioning. The macrovariables are represented by the projections Pα onthe respective Hα, where

PαPβ = δα,βPα, and∑

α

Pα = 1.We write dα for the dimension of Hα,

∑

α dα = d = dim(H); the quantity dαis the analogue of the Liouville volume |M | (1.8) of the classical case.

We also assume that the macrostates α, corresponding to the decompositionof the finite-dimensional Hilbert space H (1.20) are mapped into each othervia the involution π, i.e.,

πPαπ = Pα′ ≡ Pπα (1.21)for some α′, for each α and we write πα = α′. Remark that this implies

dπα = dα,

just like in the classical case (1.12).

1.3.4 Pure states and statistical mixtures

Up to now, we have discussed quantum systems for which the microscopicstate |ψ〉 of the entire system was well known. In many practical situations,this is of course not the case: if we are given the temperature of the system, wehave knowledge about the average energy of all the particles, and with someluck we can derive more information about the actual distribution of energiesof the system.

The question is now how to incorporate our incomplete knowledge of the mi-croscopic state of the system into the quantum formalism. A natural way todo this, is by using probability theory and introducing statistical mixtures ofquantum states.


The density matrix

Suppose that the states |ψk〉 form a complete, orthonormal set with9 k ∈{1, . . . , d}. Write Pk = |ψk〉〈ψk| the projector on the state |ψk〉, then theprojectors Pk form an orthonormal decomposition of unity:

d∑

k=1

Pk = 1, PkPm = δkmPk.We are given a probability distribution pk on the states |ψk〉,

∑

k pk = 1. If weknow that the system is in state |ψk〉 with a probability pk, then the densitymatrix ρ, describing this statistical mixture of quantum states, is defined as

ρ =

d∑

k=1

pkPk =

d∑

k=1

pk |ψk〉〈ψk| . (1.22)

Due to the lack of exact information about the microscopic state of the system,we simply refer to ρ as the state of the quantum system under consideration.

A state ρ is called a pure state if the probability distribution pk concentrateson one value of k, i.e.,

ρ = |ψk〉〈ψk| .We find that for pure states, ρ2 = ρ. It can be proven that this is a necessaryand sufficient condition for a state to be pure. This also illustrates how pro-jection operators can be used to describe systems for which the expectationvalue of a given observable is known.

It is important to emphasize that we now have two different sources of stochas-ticity in the quantum description:

• First of all, there is the probability in the initial distribution of the sys-tem. This is the stochasticity that enters through the probability distri-bution pk of above. This expresses our ignorance of the exact microscopicstate: the quantum system is in the pure state |ψ1〉 with probability p1or it is in the pure state |ψ2〉 with probability p2 or . . . or it is in thepure state |ψd〉 with probability pd.

• Secondly, there is the fundamental probabilistic aspect of quantum me-chanics which is hidden in the collapse postulate: a wave function |ψ〉can be a superposition of several eigenstates of a given observable, i.e.,

|ψ〉 =d∑

k=1

ck |ψk〉 .

9We will assume that there is no degeneracy in the eigenvalues of the eigenstates |ψk〉,so the index k runs from 1 to d, the dimension of the Hilbert space. In case of degeneracy,this upper index should be appropriately adjusted in all the formulas.


This means that the system is in the pure state |ψ1〉 with a probability|c1|2 to find it there when measured and that the system is in the purestate |ψ2〉 and . . . and in the pure state |ψd〉. Related to this is theprobabilistic nature of the measurement, which was discussed at the endof section 1.3.2.

Properties of the density matrix

We quickly review the most important properties of the density matrix. Thedensity matrix is a positive (and thus Hermitian) operator with trace one:

ρ > 0, Tr[ρ] = 1.

The expectation value of an observable A in the state described by the densitymatrix ρ is given by

〈A〉ρ = Tr[ρA] =d∑

k=1

pk 〈ψk|A|ψk〉 . (1.23)

In other words, 〈A〉ρ is the pk-weighted average of the expectation values of Ain each pure state |ψk〉 of the mixture ρ.

The time evolution as described by the Schrödinger equation can be lifted tothe level of density matrices. The resulting equation is the quantum analogueof the Liouville equation (1.6):

dρtdt

= − i~[H, ρt],

where H is the Hamiltonian describing the system. The formal solution of thisequation is given by

ρt = exp

(

− i~[H, ·]

)

ρ0,

which is very similar to equation (1.15) for pure states. Using the notation Utfor the unitary evolution (1.15), we get

ρt = Utρ0U†t , (1.24)

which should be compared to the evolution (1.16) for observables.

For composite quantum systems, the partial trace can be used to compute themarginal density matrices for a component of the system. For a Hilbert spaceH = H1⊗H2, the partial trace operation for the first component H1 is definedas

Tr1[A⊗B] = ATr[1⊗ B]which can be linearly extended to general operators on H. At the level ofdensity matrices, we write

ρj = Trj ρ (1.25)

for the marginal at component j of the density matrix ρ in some decompositionH = H1 ⊗ . . .Hj ⊗ . . .⊗Hn of the full Hilbert space.


Density matrices and measurements

Write the spectral decomposition of an observable A as

A =

d∑

j=1

ajPaj ,

where the Paj are orthogonal projectors. Given a general density matrix ρ, wecan take A = Paj in the expectation value (1.23) to find the generalization ofequation (1.19):

Probρ(A = aj) ≡ Tr[Paj ρ] =d∑

k=1

pk 〈ψk|Paj |ψk〉 , (1.26)

which gives the probability for the value aj to be measured when the state ofthe system is given by ρ.

Whenever a measurement occurs, the collapse of the wave function is imple-mented by a projection corresponding to the after-measurement eigenspace,see equation (1.18). When the measurement on a system described by densitymatrix ρ has the specific outcome aj , then von Neumann [125, p.351] teachesthat the corresponding after-measurement density matrix is given by

ρ′ =PajρPaj

Tr[PajρPaj ], (1.27)

i.e., we sandwich the original density matrix between the projector correspond-ing to the outcome and normalize this new density matrix. We can also mea-sure the system without looking at the outcome: in this case, we don’t collapseonto one particular realization of the observable A:

ρ′ =

d∑

j=1

PajρPaj (1.28)

We skip the discussion of nonideal or other than von Neumann measure-ments, where the projectors are replaced by positive operator valued measures(POVM).

1.4 The description of stochastic systems

At many instances, our understanding of physical phenomena involves sta-tistical considerations. Even when God does not play dice and also for thedescription of the classical world, depending on the scale of the phenomena,stochastic dynamics enter. They can be the result of pure modelling or they ap-pear as an effective or reduced dynamics. Classical examples are the Langevindescription of Brownian motion, the Onsager-Machlup description of fluctu-ating hydrodynamics and the stochastically driven Navier-Stokes equation for

The description of stochastic systems 33

turbulence. Since Pauli’s work [93], stochastic processes are also obtained asthe result of quantum considerations, for example also under situations wherethe Golden Rule applies. In fact, a great deal of the so-called classical stochas-tic processes, like Glauber and Kawasaki dynamics, find their origin in studiesof quantum relaxation processes. Moreover, practically all quantum processesfor open systems that are obtained under the weak coupling limit are juststandard Markov processes.

1.4.1 Detailed balance

We start by explaining what is a time-reversible stochastic process.

Restricting ourselves to stationary Markov processes, the law of the dynamicsis given in terms of transition probabilities p(x, s → x′, t) to find x′ at timet, when the system is in state x at time s. To make it simple, we supposediscrete time and a finite state space. We thus have a sequence or trajectoryX0, X1, . . . , Xt, . . . of variables as sampled from a probability law that specifies

Prob[Xt = x] = ρ(x),

the stationary distribution and the transition probabilities

Prob[Xt = x′|Xt−1 = x,Xt−2 = a2, . . . , X0 = at] ≡ p(x→ x′).

We look at that stochastic dynamics in the time-window [0, τ ] but runningthe movie backwards10, i.e., in terms of Yt ≡ Xτ−t. Obviously, the time-reversed process is also Markovian and with the same stationary probabilitydistribution ρ:

Prob[Yt = y] = Prob[Xτ−t = y] = ρ(y).

Its conditional probabilities are obtained (not: defined) from Bayes’ rule:

Prob[Yt = y′|Yt−1 = y] = Prob[Xτ−t+1 = y|Xτ−t = y′]

Prob[Xτ−t = y′]

Prob[Xτ−t+1 = y]

= p(y′ → y) ρ(y′)

ρ(y). (1.29)

As a consequence, the reversed movie will be subject to the same statistical lawwhenever and only if the transition probabilities for the time-reversed processabove

p(y′ → y) ρ(y′)

ρ(y)= p(y → y′)

equal that of the original process (right-hand side).

10For simplicity, we assume that the kinematical time-reversal π is the identity, or thatthe variable X is even under time-reversal.


Definition 1.4.1A Markov process, defined by the transition rates p(y, y′) is said to be in de-tailed balance with respect to ρ if and only if there exists a probability measureρ(y) such that

p(y → y′)p(y′ → y) =

ρ(y′)

ρ(y). (1.30)

That is certainly the case when the transition probabilities take the form

p(y → y′) = Φ(y, y′) e[V (y)−V (y′)]/2, Φ(y, y′) = Φ(y′, y),

and then ρ(y) = e−V (y)/Z is stationary. In this case, one gets for the detailedbalance condition

p(y → y′)p(y′ → y) = e

V (y)−V (y′). (1.31)

1.4.2 Continuous diffusion processes

The idea of a Markov process can be generalized to continuous phase spaces.The most famous example is without doubt the Brownian motion which rep-resents a diffusion process on, e.g., the real line: at every instance of time(whatever that means), the Brownian particle throws a coin and decides to goeither left or right, independent of its history.

Langevin dynamics

A larger and more interesting class of dynamics is given by the Langevinequation.11 This dynamics simulates a particle with mass m that moves underthe influence of a force F = −∇U , and is subjected to a frictional force Ff =−γpt/m. At the same time, the particle is in contact with a thermal reservoirat inverse temperature β = 1/kBT . The effect of the thermal reservoir onthe particle is that sometimes, its momentum is changed due to the exchangeof thermal energy with the reservoir. Writing xt = x(t) for the position andp(t) = pt for the momentum, this dynamics can be modelled by equations ofthe form

dxtdt

=ptm,

dptdt

= −∇U − γmpt +

√2m

βξ(t),

(1.32)

where the random force ξ(t) represents the “kicks” that the particle gets everynow and then from the reservoir. In the absence of the reservoir, this dynamicsreduces to the standard Newtonian dynamics.

If we do not know anything about the nature of the random force ξ(t), it iscommon to assume that it is a so-called Gaussian white noise random process.

11We follow the excellent discussion of Mortensen [89] as a nontechnical introduction tostochastic calculus.


This means that the random variable should be Gaussian distributed withmean zero (on the average, the reservoir has no influence) and has no temporalcorrelation (white noise). We get:E(ξ(t)) = 0, E(ξ(t)ξ(s)) = δ(t− s).This last equation contains a Dirac delta-function, and this indicates thatproblems will arise when one tries to solve the differential equations of mo-tion. Just as the delta-function behaves problematic when one tries to dononlinear operations12, it turns out that the Gaussian white noise becomesequally problematic to handle rigorously.

In order to avoid the mathematical pathologies associated to white noise, usu-ally its integral is used instead:

Wt =

∫ t

0

dτ ξ(τ).

The random variable Wt is a so-called Wiener or Brownian motion process,which can be defined independently of ξ(t) by saying that:

Wt ∼ N(0, t), E(Wt) = 0, E(WtWs) = min{t, s}.In a very first and heuristic interpretation13, we can read ξ(t) as dWt/dt,the derivative of the Brownian motion. By multiplying the equation for themomentum by dt, we get

dpt = −∇Udt−γ

mptdt+

√2m

βdWt.

For reduced descriptions, one often forgets the momentum and considers onlythe position. When there is no net acceleration of the particle, then of coursedpt = 0. This simplification is also referred to as the overdamped case. Substi-tuting this, together with γptdt = γmdxt, in the general form of the dynamics(1.32) and putting γ = m = 1 for simplicity, we get:

dxt = F (xt, t)dt+

√2

βdWt. (1.33)

This stochastic differential equation is commonly referred to as the Langevinequation of motion for particles under influence of a force F (xt, t) = −∇U anda thermal reservoir at inverse temperature β. The analogy with a Brownianparticle in a fluid should be clear: if there is no force F (xt, t) acting on theparticle, the equation reduces to one describing a pure Brownian motion.

12One should not try to make too much sense out of equations like δ2(x) ln δ(x), evenunder an integral.

13Or rather: mathematically incorrect. The path of a Brownian particle is not differen-tiable almost everywhere.


The Fokker-Planck equation

The Langevin equation (1.33) describes the motion of a single particle and isthe stochastic analogue of Newton’s equation, or the Hamilton equations (1.1).In order to know how an initial distribution of particles evolves under theHamilton equations, we lift these equations to the level of distributions, whichgives rise to the Liouville equation. If we do the same for the Langevin equa-tion, we obtain the Fokker-Planck equation. The distribution ρ(x, t) evolvesaccording to

∂ρ(x, t)

∂t= − ∂

∂x

(

F (x, t)ρ(x, t)

)

+1

β

∂2ρ(x, t)

∂x2.

The influence of the reservoir is visible in the Laplacian on the right: thedistribution is smeared out with a diffusion constant proportional to the tem-perature. A special case corresponds to F (x, t) = −∇U(x) for some potentialU , which should be thought of as the analogue of the detailed balance condi-tion (1.31), but we give no further details. Then, the “stationary” distributionfor this equation, in the sense that

∂ρ

∂t= 0,

is the Gibbs distribution

ρ(x, t) =e−βU(x,t)

Zt, Zt =

∫ ∞

−∞

dx e−βU(x,t). (1.34)

A general note on stochastic differential equations

The Langevin equation (1.33) can be cast in a more general form as

dxt = F (xt, t)dt+ g(t) dWt,

which in turn can be written as the integral equation

xt = x0 +

∫ t

0

dτ F (x, τ) +

∫ t

0

dWτ g(τ).

The integral∫dWτ g(τ) is with respect to a Brownian process and is a new

kind of integral, the so-called stochastic integral. It turns out that there aretwo ways of interpreting this integral [81, 89]:

• The interpretation of Itô, who defined new rules for integration (which wewill not list here). The Itô integral is defined by splitting the integrationdomain [0, τ ] into a set of discrete points 0 = t0 < t1 < . . . < tn−1 <tn = τ with spacing ∆t = tj − tj−1. We evaluate the function g(t) at theleft endpoint of each interval [tj−1, tj ]:

∫ t

0

dWτ g(τ) = limn→∞

∆t→0

n∑

j=1

g(tj−1)(Wtj −Wtj−1 ),


where the limit should be understood in L2-sense. General integrals canbe reduced to this form by using the Itô rule for differentials:

dIF (Wt) = F′(Wt)dWt +

1

2F ′′(Wt)dt,

where dI means Itô differential.

• The interpretation of Stratonovich, who used the Itô definition of thestochastic integral and extended it (with limited validity) in order tofind back the standard rules of (Riemann) integration. The Stratonovichintegral is indicated by a circle “◦” to emphasize the difference with theItô integral. The Stratonovich integral is also defined by splitting theintegration domain [0, τ ]. In contrast with the Itô definition, now weevaluate the function g(t) at the middle of the intervals [tj−1, tj ] to get:∫ t

0

dWτ ◦ g(τ) = limn→∞

∆t→0

n∑

j=1

g

(tj−1 + tj

2

)

(Wtj −Wtj−1 )

= limn→∞

∆t→0

1

2

n∑

j=1

(g(tj) + g(tj−1)

)(Wtj −Wtj−1 ), (1.35)

which shows precisely where the deviations from the Itô definition orig-inate from. General integrals can be reduced with the usual rule forintegration:

dSF (Wt) = F′(Wt)dWt,

where dS stands for Stratonovich differential.

Both interpretations give different results for the integrals, but they can berelated using a transformation formula. The Itô integral remains more funda-mental and more general than the Stratonovich integral.

In this work, we will use the Stratonovich interpretation, for reasons that willappear evident in chapter 6.

1.4.3 Elements of stochastic calculus

To conclude the section on stochastic descriptions, we review some convenienttools for the manipulation of probability distributions. It is not our intentionto be entirely rigorous, we refer to the literature [76] for a full mathematicaltreatment of these subjects.

The Radon-Nikodým derivative

Suppose we are given a probability density ρ(x) and µ(x) on R.Definition 1.4.2 (Absolutely continuous)We say that the density ρ is absolutely continuous with respect to the densityµ, written ρ≪ µ, if

µ(A) = 0 ⇒ ρ(A) = 0, ∀A ⊂ R.


We can ask if it is possible to relate expectation values under one probabil-ity distribution with the other. This is precisely the content of the Radon-Nikodým theorem14

Theorem 1.4.1 (Radon-Nikodým)If ρ≪ µ, then there exists a measurable function f : R→ R+ such that

ρ(A) =

∫

A

fdµ.

We write the Radon-Nikodým derivative f as

f =dρ

dµ.

We will often work with measures ρ which are absolutely continuous with re-spect to the Lebesgue measure “dx”. In this case, we can write dρ = ρ(x)dxwhere the Radon-Nikodým derivative ρ(x) is called the density function. Some-times, we will write the Radon-Nikodým derivative in terms of the densityfunctions ρ(x) and µ(x) instead of using the probability densities dρ and dµ,i.e., “dividing out” the Lebesgue measure dx, we get

f =dρ

dµ=ρ(x)

µ(x), (1.36)

where dρ = ρ(x)dx and dµ = µ(x)dx. In this sense, the Radon-Nikodýmderivative is a ratio of density functions.

The Radon-Nikodým derivative has all the usual properties of a good deriva-tive:

• If ρ≪ µ and ν ≪ µ, thend(ρ+ ν)

dµ=dρ

dµ+dν

dµ, µ− a.e.

• If ρ≪ µ≪ λ, thendρ

dλ=dρ

dµ

dµ

dλ, µ− a.e.

• If ρ≪ µ and µ≪ ρ, then

dρ

dµ=

(dµ

dρ

)−1

.

• If ρ≪ µ, then for integrable functions g we have

Time, Heat and Work · 2010. 3. 29. · Katholieke Universiteit Leuven Faculteit Wetenschappen...

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Transcript of Time, Heat and Work · 2010. 3. 29. · Katholieke Universiteit Leuven Faculteit Wetenschappen...