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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Authored by a well-known expert in the field of nonequilibrium statistical physics, this book is a coherent presentation of the subject suitable for masters and PhD students, as well as postdocs in physics and related disciplines.
Starting from a general discussion of irreversibility and entropy, the method of nonequilibrium statistical operator is presented as a general concept. Stochastic processes are introduced as a necessary prerequisite to describe the evolution of a nonequilibrium state. Different standard approaches such as master equations, kinetic equations and linear response theory, are derived after special assumptions. This allows for an insight into the problems of nonequilibrium physics, a discussion of the limits of the approaches, and suggestions for improvements. The method of thermodynamic Green's function is outlined that allows for the systematic quantum statistical treatment of many-body systems. Applications and typical examples are given, as well as fully worked problems.
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Seitenzahl: 561
Veröffentlichungsjahr: 2013
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Contents
Cover
Related Titles
Title Page
Copyright
Dedication
Preface
Chapter 1: Introduction
1.1 Irreversibility: The Arrow of Time
1.2 Thermodynamics of Irreversible Processes
Exercises
Chapter 2: Stochastic Processes
2.1 Stochastic Processes with Discrete Event Times
2.2 Birth-and-Death Processes and Master Equation
2.3 Brownian Motion and Langevin Equation
Exercises
Chapter 3: Quantum Master Equation
3.1 Derivation of the Quantum Master Equation
3.2 Properties of the Quantum Master Equation and Examples
Exercises
Chapter 4: Kinetic Theory
4.1 The Boltzmann Equation
4.2 Solutions of the Boltzmann Equation
4.3 The Vlasov–Landau Equation and Hydrodynamic Equations
Exercises
Chapter 5: Linear Response Theory
5.1 Linear Response Theory and Generalized Fluctuation–Dissipation Theorem (FDT)
5.2 Generalized Linear Response Approaches
Exercises
Chapter 6: Quantum Statistical Methods
6.1 Perturbation Theory for Many-Particle Systems
6.2 Thermodynamic Green's Functions
6.3 Partial Summation and Many-Particle Phenomena
6.4 Path Integrals
Exercises
Chapter 7: Outlook: Nonequilibrium Evolution and Stochastic Processes
7.1 Stochastic Models for Quantum Evolution
7.2 Examples
References
Index
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Dedicated to Friedrich, Franziska, Johanna, and Heide-Luise
Preface
All men are like grass,
and all their glory is like the flowers of the field;
the grass withers and the flowers fall,
and its place remembers it no more.
– Moses, Psalm 90: 5–6; David, Psalm 103: 15–16; Isaiah 40: 6–8; 1 Peter 1: 24–25; J. Brahms, Requiem
Irreversibility is one of the largest mysteries of science at the present time. Birth and death, creation, evolution, and destruction are fundamental human experiences. We feel the arrow of time that determines past, present, and future. We measure time with nearly reversible, periodic processes, but there is also another aspect of time that is related to irreversible changes. It is a challenge to give a consistent approach to general nonequilibrium phenomena. Do we need new concepts and new mathematics in this context?
Nonequilibrium physics concerns different phenomena such as evolution, relaxation to equilibrium, friction, and other transport phenomena. In addition, we wish to consider the reaction of a system to external influences, the role of fluctuations, metastability and instability, pattern formation and self-organization, the role of probability and chance in contrast to a deterministic description, and the treatment of open systems. Statistical physics of nonequilibrium has created some concepts and models that are of relevance not only to physics but also to other fields such as informatics, technology, biology, medical, and social sciences. It also has an impact on fundamental philosophical questions. The treatment of nonequilibrium phenomena is an emerging field in physics and is of relevance to other fields such as quantum physics and field theories, phase transitions, bio- and nanophysics, and evolution of complex systems.
A central point is thermodynamics that introduced a new quantity, the entropy, not known in the other disciplines of theoretical physics. The second law of thermodynamics states that the entropy in an isolated system can increase but never decrease with time. Up to now, a consistent “first principle” theory of irreversible processes based on the fundamental, but reversible, equations of motion of microscopic dynamics is missing. To move toward an explanation of irreversible phenomena, we have to inquire into some paradigms used in the present-day physics, for example, the complete separation of a system from its surroundings.
In contrast to equilibrium statistical physics, nonequilibrium statistical physics is only rarely part of current courses in theoretical physics. We are at present not able to formulate axioms or principles that allow a general approach to describe nonequilibrium physics. Only for special situations, we know different approaches that can be used to describe properties of a nonequilibrium process. In all cases, we have to add some assumptions or approximations that seem at first glance to be an inaccuracy within the strict microscopic treatment, but, on the other hand, bring a new element into the theory that seems to be indispensable to describe irreversible behavior.
A first microscopic approach to irreversible processes was given by Ludwig Boltzmann in 1873 investigating the kinetic theory of gases. The Boltzmann equation [1] that remains as a basic equation until now is based on the equations of motions for atomic collisions, but needs an additional element, the “Stoßzahlansatz” or the molecular chaos. This way, the famous H theorem explicitly shows the selection of the direction of time and the possibility to describe irreversible evolutions, starting from reversible equations of motion that describe the microscopic dynamics of the molecules.
A more systematic derivation of the Boltzmann equation was given in 1946 by Bogoliubov [2] using the principle of weakening of initial correlations. To begin with many-particle systems at low density described by the single-particle distribution function, quantum statistical methods such as the time-dependent Green's function technique [3] have been worked out to treat also systems at higher densities. Theories for transport processes in dense systems are formulated such as the linear response theory by Kubo [4], which relates the dissipation of a nonequilibrium initial state to the evolution of fluctuations in the equilibrium system, for instance, the conductivity to current–current correlation functions.
Another approach was the projection operator technique by Nakajima and Zwanzig [5] that allowed deriving an irreversible equation, the Pauli equation, from the microscopic von Neumann equation of motion for the statistical operator. The additional assumption was that the nondiagonal elements of the density matrix are fading. This approach has been developed further to describe relaxation processes. It is presently considered in relation to decoherence and the physics of open systems.
Different nonequilibrium phenomena are described by the respective theories. The assumptions made in addition to solving the microscopic equations of motions are reasonable for the case under consideration. We have detailed monographs for different fields. As examples, the thermodynamics of irreversible processes [6], the kinetic theory [7], the linear response theory [8], different approaches in the series of Landau and Lifshits [9,10], and the theory of open systems [11,12] should be mentioned. All these approaches use some additional assumption that introduces a reduced set of relevant observables. A unified approach was given with the Zubarev method of the nonequilibrium statistical operator [13].
This book intends to give a coherent, concise, general, and systematic approach to different nonequilibrium processes. The main point of Chapter 1 is to state the problem. After discussing some basics of other cognate disciplines in theoretical physics, empirical approaches are explained. Stochastic processes like the Langevin process or random walk that are characteristic of nonequilibrium behavior are introduced in Chapter 2. Three typical domains – quantum master equations (Chapter 3), kinetic theory (Chapter 4), and linear response theory (Chapter 5) – are presented in detail. Examples are given, in particular, the radioactive decay described by a Pauli equation and the electrical conductivity in charged particle systems. Quantum statistical methods to treat many-particle systems are given in Chapter 6, concluding with an outlook in Chapter 7.
The book should make nonequilibrium statistical physics accessible to students and scientists interested or working in that field. For an extended presentation and advanced examples, refer to Refs. [14,15]. We will not divide between classical and quantum physics, but consider classical physics as a limiting case of quantum physics.1 We focus on applications in solid-state physics, plasma physics, subatomic physics, and other fields where correlations are of relevance to many-particle systems. Other interesting fields, like nonequilibrium QED, phase transitions, measuring process, cosmology, turbulence, relativistic systems, and decoherence, are only briefly mentioned or even dropped.
Based on lectures given at Dresden, Rostock, Greifswald, and other places, a previous textbook was published in German [17], thanks for help in preparation to Heidi Wegener, David Blaschke, Fred Reinholz, and Frank Schweitzer. In Ref. [17], solutions are found for some problems given in the present book. A translation to Russian [18] was performed by Sergey Tischtshenko. The Green's function method was worked out as a script material by Holger Stein and improved by Mathias Winkel that served as prototype of Chapter 6. Also, the nonequilibrium statistical physics script was worked out further with the help of Jürn Schmelzer (Jr.), Robert Thiele, Thomas Millat, Carsten Fortmann, and Philipp Sperling. A lot of discussions have been performed on this subject in Rostock, Moscow, and other places. We are grateful to Dmitri Zubarev, who made me familiar with nonequilibrium thermodynamics during my postdoc stay at the Steklov Mathematical Institute of the Soviet Academy of Science, Moscow, in 1969. We also acknowledge Vladimir Morozov, Ronald Redmer, Heidi Reinholz, Werner Ebeling, Wolf-Dieter Kraeft, Dietrich Kremp, Klaus Kilimann, David Blaschke, Michael Bonitz, Thomas Bornath, Sibylle Günter, Claudia-Veronika Meister, Klaus Morawetz, Manfred Schlanges, Sebastian Schmidt, August Wierling, and others who developed quantum statistics and nonequilibrium processes during the last decades at Rostock.
Rostock, October 2012
Gerd Röpke
Note
1. Note that the appearance of the classical world from quantum theory is not trivial and has to be analyzed within nonequilibrium physics [16].
1
Introduction
Physics is concerned with phenomena in nature. It describes properties of systems and their time evolution. Very efficient concepts have been worked out, and detailed knowledge about nature has been accumulated. A lot of phenomena can be explained using very few basic relations. However, there also exist unsolved problems. Such a field with open questions is the physics of nonequilibrium processes, where until now no fundamental and coherent approach has been possible.
Nonequilibrium is the general situation in the real world. Change in time is one of our direct experiences; , everything flows, one does not step into the same river twice, as was pointed out by the ancient philosophers [19].
We experience dissolution, destruction, formation of new structures, higher complexity, and higher organization; possibly we believe in progress, everything is going to be alright. Evolution in biological (and social) systems is a great miracle.
Can we understand the evolution of a system and even predict the future? Why are we interested in the future? To avert danger, to optimize our situation, to realize our goals, and to see what remains. We have to make decisions and anticipate the consequences.
Physics contributes a lot by analyzing the dynamical behavior of matter. A deterministic description based on the solution of the fundamental equations of motion was promoted by its success in celestial dynamics. This formed our present approach to describe phenomena by equations of motions that have the form of differential equations. We present some fundamental equations and show that they describe reversible dynamics. Consequently, an “arrow of time” does not exist here, as detailed in Section 1.1.
The paradigm of the deterministic description is well characterized by the so-called “Laplace intelligence”: “Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom; to it nothing would be uncertain, and the future as the past would be present to its eyes” [20].
The great success in using our fundamental equations of motions to describe all observed phenomena convinced people to believe in a deterministic approach. The exact predictability of the future, however, seems to be an illusion because of different reasons as discussed later on. In contrast to exact predictability, we introduce in Chapter 2 a probabilistic description.
We present in Section 1.2 some ideas that may contribute to the solution of the problems associated with the contradiction between irreversible evolution and reversible dynamics. We point out that the assumptions made in formulating the dynamics on the basis of the equations of motion have to be critically analyzed.
One of the basic ideas is the assumption that a system can be separated from the remaining part of the universe. Its time evolution can be described taking into account the influence of the surroundings via simple approximations.
As an example, the equations describing the motion of planets can be given neglecting the influence of astronomical objects outside the solar system. Furthermore, all the complex processes that take place on each planet are neglected. Only the center of mass motion is considered.
With respect to the motion of the planets, it is sufficient to consider only a restricted number of relevant observables characterizing the state of the system. Other observables, for example, those related to the internal state of the object, are irrelevant.
The number of relevant observables describing the state of the system is given by the degrees of freedom. One has to distinguish between the dynamical degrees of freedom, which are available by the motion of the system, and the constraints, which reduce the number of dynamical degrees of freedom.
A perfect isolation of a system from the remaining part of the universe is not possible. For example, Mach's principle relates the motion of the distant stars to the local inertial frame. It is currently not possible to disconnect gravity. As a consequence, each system also “feels” the expansion of the universe.
To make statements precise, we will give relevant results obtained in other fields of physics, known from standard courses, without extended derivations. A detailed discussion of some of the relations presented here is given later on. The corresponding references are given in the text.
1.1 Irreversibility: The Arrow of Time
We are concerned here only with “dead” matter, particles, and their interactions. The behavior of such systems is described by “microscopic,” dynamical equations of motion. Examples are the Newton equation, the Schrödinger equation, the Maxwell equations, and quantum electrodynamics.
We give some standard results and briefly show some general results from other fields of theoretical physics. Then, we discuss the irreversible “macroscopic” evolution of real systems. A detailed discussion of the equations of motion in different fields of physics is given later on. We focus only on properties with respect to time reflection, showing that there is no difference between past and future, no “arrow of time”; a reverse motion picture would also show a possible solution of the equations of motion, a possible microscopic process.
Thermodynamics [21] is a phenomenological theory, directly related to quantities that can be measured. A well-known fact is that the second law singles out an “arrow of time.” In an isolated system, the entropy will increase with time for the evolution of any nonequilibrium state. If we make a motion picture for a real phenomenon (not only friction and diffusion but also living creatures), a reverse motion picture would not show a possible real phenomenon.
As a typical example, we consider a many-particle system. The microscopic equations of motion follow from a Hamiltonian, for example, Newton's equation of motion in classical physics. For quantum systems, second quantization is very convenient to calculate properties. We also consider the statistical operator that gives a link between phenomenological properties and the microscopic dynamics. We focus here on properties with respect to time reflection and show that the equation of motion for the statistical operator, obtained from the Schrödinger equation, cannot describe irreversible processes. The appearance of the “arrow of time” in real phenomena [22] is a mystery in our present fundamental understanding of time evolution of a system using a microscopic approach.
1.1.1 Dynamical Systems
The state of a system at fixed time is characterized by a number of variables, the values of which can change with time. This number , the degrees of freedom, may be finite. For example, in thermodynamics of compression processes and chemical reactions, and the variables are the volume , the particle number , and the temperature .1 We are concerned in the following with a system of point masses in classical mechanics where for the Cartesian components of the position and momentum vectors. It may also be infinite, for example, for fields (electrical field, state vector in quantum mechanics, etc.) where for each of an infinite number of positions in space, the corresponding value of the field must be known. Alternatively, we can characterize a field by components with respect to a (infinite) basis system of orthonormal functions.
Classical Mechanics
Can we predict the future of the state of the system if we know its initial state, that is, can we predict the change in the values of the state variables with time? As an example, we can consider the system of point-like interacting particles as an idealization used in celestial mechanics2 or in molecular dynamics. The state in configuration space has degrees of freedom. The forces are assumed to be conservative. To solve the equations of motion that define a special trajectory , for example, to solve the Newton equations or the Hamilton equations, we need initial conditions. The actual state in configuration space is not sufficient, we also need to have the information about the actual values of the velocities or the canonical conjugate momenta. To determine the dynamical state of the system, we have to extend the set of state variables (degrees of freedom), that is, the configuration space to the dimensional space that also includes, besides the positions, the particles' momenta.
For a classical system of particles, the dynamics is determined by the Hamilton function, which is the sum of kinetic and potential energy:
(1.1)
where denotes the particle number. In general, the external potential can be time dependent, , for example, charged particles in a time-dependent electrical field. The interaction potential is given by the (conservative) forces between the particles.
The Hamilton equations
(1.2)
are first-order differential equations in time. The trajectory of the body system is determined by an initial state that is a given point in the space. The corresponding dynamics is reversible, that is, with time inversion at , we construct a new trajectory . This “new” trajectory is also a solution of the Hamilton equations (1.2) and therefore describes a possible motion.
Quantum Mechanics
A similar situation arises in quantum mechanics. The state of a single particle is given by a complete set of simultaneously measurable quantities. For instance, in the case of electron,3 we need four items of data, for example, three for the position in coordinate space () and one for the spin orientation (). In general, the state of a particle is given by the state vector. It can be represented by components in different basis systems, for example, the state function for the electron.4 Unitary transformations relate different representations, in particular the Fourier transform for the momentum representation.
The time dependence of a quantum state is determined by the Schrödinger equation
(1.4)
and an initial state . The corresponding dynamics is reversible, that is, the dynamics with time inversion also describes a possible motion, if the Hamiltonian is Hermitian. We mention that time inversion also means the adjoint complex in addition to inversion of the spin and the magnetic field.
Quantum Many-Particle Systems
The state of a quantum many-particle system is characterized by a corresponding high number of degrees of freedom. In general, the particle numbers of species are not fixed (emission and absorption of photons, open systems that are defined by a given volume in space allowing particle exchange with a reservoir, chemical reactions, phase transitions, etc.), so we can use in quantum physics the Fock space, that is, the direct sum of Hilbert spaces with arbitrary particle numbers. A convenient possibility to characterize the state of a system with arbitrary particle numbers is the occupation number representation (second quantization) where the number of particles in each single-particle state is used. The basis of the Fock space is given by the occupation numbers of the different single-particle states. Creation () and annihilation operators () are introduced that can be used to construct the basis of the Fock space and the matrix elements of any dynamical observable. The commutation or anticommutation relations are
(1.9)
for bosons, and
(1.10)
for fermions, respectively.
The Hamiltonian of a many-particle system with interaction (matrix element with respect to the single-particle states ) is
(1.11)
(the variable “species” c also contains the spin orientation. It can be included in the single-particle quantum number p). The many-particle Hamiltonian describes the dynamical evolution of the system.
The time dependence can be transformed to the Heisenberg picture. The quantum state remains unchanged, but the dynamical operator changes with time as
(1.12)
t0 denotes the instant of time where the Heisenberg picture and the Schrödinger picture coincide. The corresponding equation of motion is
(1.13)
Similar to the Schrödinger picture, this time dependence is reversible, the equations of motion for quantum many-particle systems are invariant with respect to time inversion and complex conjugation.
Electrodynamics
The dynamics is also reversible in other fields of “microscopic” physics. In electrodynamics, the state is described by both the electrical and magnetic fields. In a relativistic description, we can introduce the four-vector field at , and the four-tensor of field strengths is derived from the four-potential. The equations of motion in electrodynamics, the Maxwell equations, also describe reversible motion. After time inversion and reversal of the magnetic field, the new process is also a solution of Maxwell's equations (Problem 1.1). The quantization of the electromagnetic fields can be performed using the formalism of second quantization mentioned above.
On a very sophisticated level, we can use quantum electrodynamics to describe particles interacting with the electromagnetic field. We can consider the action or the Lagrangian of , where the state of the system is given by the real Maxwell four-vector field and the complex Dirac spinor field , where s denotes the spinor components (Problem 1.2). As we know, many phenomena in atomic physics, molecular physics, solid-state physics, plasma physics, quantum optics, liquid-state physics, ferromagnetism, superconductivity, and so on are correctly described with this Lagrangian. In particular, we obtain the Dirac equation and the Maxwell equations within the canonical formalism.
A basic property of such microscopic equations of motion is reversibility in time. Performing a time inversion, the resulting motion also seems to be a physically possible process. There is no principal difference between past and future. Periodic processes can be used to measure the time: earth around sun, rotation of earth, pendulum, vibration of quartz, and atomic clocks (Problem 1.3).
1.1.2 Thermodynamics
The microscopic description is based on different approximations and idealizations. In particular, part of the interaction that is not of relevance is dropped. Real macroscopic systems are described phenomenologically, introducing state variables. Some of them have a simple interpretation such as the volume and the particle number of species c. Also, the energy is known from mechanics as the sum of kinetic and potential energy. More generally, we can take the Hamiltonian to calculate the energy E of a system.
Other state variables are introduced via the laws of thermodynamics that are based on experience. These laws define the temperature T, the internal energy U, and the entropy S. As a consequence, the relation
(1.14)
for reversible processes is obtained. Here, only two forms of work are considered, the volume compression work (pressure p) and the chemical work (chemical potential μc).6 Reversible processes mean quasistatic, slow changes so that at each instant of time, the system is in thermal equilibrium. The first law of thermodynamics gives the increase of internal energy .7 We identify the internal energy with the energy of a system.
According to the second law, the relation for reversible processes defines the entropy that is an extensive quantity. At the same time, the temperature is defined as integrating denominator. The absolute value of the entropy is fixed by the third law of thermodynamics. For any particular system under consideration, the entropy can be determined measuring the heat capacity:
(1.15)
if other variables like are fixed. For engineers, tables are available containing, besides other thermodynamic functions, also the entropy for different materials.
Allowing also for irreversible processes,
(1.16)
according to the second law. In particular,
(1.17)
holds for the time evolution of the entropy of closed systems. For isolated systems, no exchange of heat with a bath is possible so that . Irreversible processes define a direction (arrow) of time because time inversion means that entropy would decrease in closed systems. More generally, for with ,
(1.18)
for any process. This is forbidden according to the second law of thermodynamics.
The basis for introducing the entropy is the existence of reversible and irreversible processes [21]. Three examples are discussed that establish irreversible processes: friction that transforms mechanical work into heat (e.g., pendulum with friction), diffusion of a substance to free space (e.g., dissolution of a concentration profile in a liquid), and heat transfer from warm to cold systems. It is impossible to construct a perpetuum mobile of the second kind. There is an arrow of time, and it becomes evident that the arrow of time points from the past into the future considering processes such as friction, heat conduction, and diffusion processes. Thus, the evolution of a real, macroscopic system is in general irreversible. We can distinguish between a movie of a possible process and the time inverse movie that is not possible (Problems 1.4 and 1.5).
1.1.3 Ensembles and Probability Distribution
In thermodynamic equilibrium, a connection between macroscopic and microscopic approaches can be given in the frame of statistical physics. For this, the entropy has to be introduced into the microscopic dynamical approach, which is done via probability. Once the entropy is introduced, other quantities like temperature or chemical potentials can be deduced.
Ensembles are considered instead of a particular real system. The ensembles are determined by all realizations that are compatible with the boundary conditions, given by the values of the relevant thermodynamic variables. More precisely, a probability distribution for the microstates of the dynamical system is introduced. This probability distribution is formed in such a way that the values of the relevant variables of the thermodynamic macrostate are correctly described (consistency conditions). As in quantum mechanics, we investigate only averaged properties of the ensemble, not the individual properties of the particular real system under consideration.
For quantum systems, the microstates of the dynamical system at time are given by the state vector . We suppose a complete set of commuting observables that uniquely define the microscopic state of the system, for example, the position and -component of spin of all electrons in a system of electrons. The distribution function or statistical operator8
(1.19)
contains the probability that the macroscopic system under consideration is found in the microscopic state . The probabilities are real numbers, so is Hermitian. If we have a complete set of alternative states , the probability is normalized according to
(1.20)
For any dynamical observable , the average is given by9
(1.21)
How does the statistical operator depend on time? We start with the Schrödinger equation that describes the time dependence of the states and its conjugate complex (H†=H):
(1.22)
With
(1.23)
we obtain the von Neumann equation as the equation of motion for the statistical operator:
(1.24)
The von Neumann equation describes reversible dynamics. The equation of motion is based on the Schrödinger equation. Time inversion and conjugate complex means that both terms change the sign, since and both the Hamiltonian and the statistical operator are Hermitian (Problem 1.6).
1.1.4 Entropy in Equilibrium Systems
In thermodynamic equilibrium, the state of the system is not changing with time, . There is no dependence on . The solution of the von Neumann equation becomes trivial,
(1.25)
in thermodynamic equilibrium, and the time-independent statistical operator commutes with the Hamiltonian. We conclude that depends only on constants of motion that commute with . However, the von Neumann equation is not sufficient to determine how depends on constants of motion .10
We consider a system containing particles of species with numbers . The dynamics is described by the Hamiltonian . The thermodynamic state variables are given by the contact with the “environment” (bath or reservoir). Due to these contacts, the constants of motion can fluctuate, but equilibrium means that the average values are not changing with time. Equilibrium statistical mechanics is based on the following principle to determine the statistical operator :
Consider the functional (information entropy)11
(1.26)
for arbitrary that are consistent with the fixed conditions:
(1.27)
(normalization) and
(1.28)
(self-consistency conditions). With these conditions, we vary and determine the maximum of the information entropy for the optimal distribution . The corresponding result
(1.29)
is the equilibrium entropy of the system for given constraints , is the Boltzmann constant. The solution of this variational principle leads to the Gibbs ensembles for thermodynamic equilibrium12 (see Sections 1.1.6 and 1.2.2).
As an example, we consider an open system that is in thermal contact and particle exchange with reservoirs. The sought-after equilibrium statistical operator has to obey the given constraints normalization, , thermal contact with the bath so that
(1.29a)
and particle exchange with a reservoir so that
(1.29b)
Looking for the maximum of the information entropy functional, , with these constraints, one obtains the grand canonical distribution (see also Section 1.2.2 for derivation):
(1.30)
or
(1.31)
where we introduced explicitly the eigenvalue of the particle number operator, ν = {Nc,n} contains the particle numbers N of all species and the internal quantum number n of the excitation, are the energy eigenvalues of the eigenstates of the system Hamiltonian confined to the volume (we do not use to avoid confusion with the potential). The normalization is explicitly accounted for by the denominator (partition function). The second condition (1.29a) means that the energy of a system, which is in heat contact with a thermostat, fluctuates around an averaged value with the given density of internal energy . This condition is taken into account by the Lagrange multiplier that must be related to the temperature, a more detailed discussion leads to . Similarly, the contact with the particle reservoir fixes the particle density , introduced by the Lagrange multiplier that represents the chemical potentials.
Within the variational approach, the Lagrange parameters have to be eliminated. This leads to the equations of state , (Problem 1.7). The dependence of extensive quantities on the volume is trivial. The method to construct statistical ensembles from the maximum of entropy under given conditions, which take into account the different contacts with the surrounding bath, is well accepted in equilibrium statistical mechanics and is applied to different phenomena, including phase transitions (see Refs [9,13]).
In conclusion, in thermodynamic equilibrium, a connection between the microscopic dynamical approach and the thermodynamical approach can be given. For this, the entropy has to be introduced into the microscopic dynamical approach. This is done with the help of probability.13
Can we use this definition of equilibrium entropy for evolution in nonequilibrium processes? Time evolution of ρ, Eq. (1.24), is given by a unitary transformation that leaves the trace invariant. Thus, the entropy defined above is constant.
The equations of motion, including the Schrödinger equation and the Liouville–von Neumann equation, describe reversible processes and are not appropriate for describing irreversible processes. Therefore, the entropy concept (1.29) worked out in equilibrium statistical physics cannot be used as a fundamental approach to nonequilibrium statistical physics.
Up to now, there is no basic approach for how to extend the concept of entropy to nonequilibrium processes. There are different situations where equations of evolution can be given, which contain, in addition to the dynamical description, phenomenological concepts (e.g., the Boltzmann equation and the “Stoßzahlansatz”). In this book, we attempt the formulation of a coherent description applicable to different nonequilibrium processes. We indicate clearly where additional arguments are introduced to obtain irreversible equations of evolution that do not conflict with equilibrium descriptions.
1.1.5 Fundamental Time Arrows, Units
The problem of treating irreversible processes is connected with the arrow of time. Past, present, and future are different. Is there a common, general phenomenon that defines the arrow of time? Various processes are known where the time direction is singled out [22].
An interesting question is whether there exists a “master arrow of time” that also defines the other observed arrows of time.
The following issues are also of importance in connection with irreversibility:
Chaotic motion. The equation of motion for dynamical systems can show dynamical instabilities, so the trajectory becomes unpredictable over long time intervals. This happens in particular for complex systems. The Lyapunov exponent indicates how the distance between two neighboring points in phase space increases with time.Open systems. Basically, each system is subject to a coupling to the surroundings or to further internal degrees of freedom. This makes the time evolution not fully dynamical, but introduces an additional influence that is not under control. By this, probabilities enter the time evolution (in addition to quantum behavior such as the unpredictable decay of an excited quantum state).Stochastic processes. These give a mathematical tool to describe probabilistic behavior in the time evolution and to obtain irreversible equations of evolution.Units
To investigate the origin of irreversible behavior, it is of interest to consider appropriate units, similar to quantum mechanics that is characterized by atomic units.
It is assumed [25] that the relevant scales on which effects of quantum gravity should be unavoidable are given by the Planck length , the Planck time , and the Planck mass . These are given by the following expressions:
(1.34)
(1.35)
(1.36)
The universal gravitational constant appears in Newton's law:
(1.37)
A quantity expressing the ratio of atomic scales to the Planck scale is the “fine structure of gravity” defined by
(1.39)
where denotes the proton mass.15
A characteristic time that describes radiation damping (of electrons) is (Problems 1.8 and 1.9)
(1.41)
It appears in the Abraham–Lorentz equation [26],
(1.42)
There is also gravitational radiation. Because it is quadrupolar (graviton with spin 2), radiation losses are different. A classical equation of motion with radiation damping reads [27]
(1.43)
Here, a characteristic time occurs:
(1.44)
Orbital decay from gravitational radiation has been observed in binary systems (masses and distance ) with a relaxation time (Problem 1.10):
(1.45)
These units are of relevance for irreversible processes.16
A final comment concerns the continuum limit. It is based on the assumption that we can scale the intervals downward so that some properties remain constant. Let us consider the mass density or the charge density that are well-defined quantities for a particular material sample. However, it becomes meaningless and ill defined at atomic scales. In an atom, almost the entire mass is concentrated in the small atomic nucleus. The remaining large volume is nearly empty. The concept of density applies only to finite intervals that can be scaled down in macroscopic scales so that the density stays nearly constant. As soon as we investigate the sample on atomic scales, the concept of a density becomes meaningless.
We will also consider time derivatives that may become problematic at small scales, for example, the Planck time. We can consider differences at discrete time intervals and scale it downward. However, we should be aware that this limiting process becomes meaningless below some typical units where new physics appears.
We can consider the Planck units as an ultimate limit to apply our present knowledge about the structure of space and time. The use of the concept of a continuum expresses only the invariance of some results with respect to a finer scale so that a limit process can be performed. But we know that this limit does not exist, so we are always confined to work with discrete structures in space and time.
1.1.6 Example: Ideal Quantum Gases
As an example, we consider at first a system of noninteracting particles in classical mechanics. The Hamiltonian contains only the kinetic energy:
(1.46)
This model can be exactly solved; there are dynamical degrees of freedom and conserved quantities, the three components of each particle's momentum and initial position . All particles are moving independently along the straight lines:
(1.47)
To be a well-defined thermodynamic system, we have to introduce a volume so that we can also introduce a density . This can be achieved by perfectly reflecting walls that confine the system. An alternative is periodic boundary conditions.16a
The entropy for the ideal classical monoatomic gas (point masses without internal degrees of freedom) is obtained, for example, from the free energy by evaluating the canonical partition function (Problem 1.12):
(1.48)
Ideal quantum gases are commonly investigated in the grand canonical ensemble. Using the occupation number representation, we have
(1.49)
We find from the partition function the grand canonical potential so that
(1.50)
with the upper sign for fermions and the lower sign for bosons. Differentiating this expression with respect to , we obtain the well-known Fermi and Bose distribution functions:
(1.51)
In terms of the distribution function, for the particle number we find ; furthermore, and
(1.52)
(1.53)
These expressions can be discussed in terms of the occupation of particle and hole () states.
We will not discuss here the questions of the low-temperature limit of Fermi gases where the ground state can be considered as the occupied Fermi sphere (Dirac sea). The excitations above the Fermi energy are described as particle states and the empty states as hole states. In Bose systems, at low temperatures, Bose–Einstein condensation may occur. These phenomena are well described in the standard textbooks of statistical physics of equilibrium. For Bose systems, as an alternative to the occupation number representation, coherent states can also be introduced. The relation to classical physics, the Wigner function, the Glauber states, and so on, are discussed later on (Problems 1.13 and 1.14).
A difficulty in ideal quantum gases is that there is no interaction process for any nonequilibrium state to go to equilibrium. For light in a perfectly reflecting hohlraum, for example, we need a (“infinitesimal”) small interaction process to reach thermal equilibrium. The famous “Planck'sche Staubkorn” (dusty particle) in a “hohlraum” is necessary to absorb and emit light.
1.2 Thermodynamics of Irreversible Processes
In the previous section, we discussed microscopic descriptions. This implies a system separated from the universe so that the state is described by a small number of degrees of freedom. Let us consider, for example, present models for physical processes on the atomic scale. Solving the Schrödinger equation, we obtain stationary states with infinite lifetime for excited atomic states. The microscopic theory gives very accurate results, but obeys reversible dynamics. In contrast to microscopic physics, thermodynamics is a phenomenological approach. It describes real matter, not microscopic models, and contains the phenomenon of irreversibility.
The microscopic explanation of irreversibility is an unsolved problem. Therefore, it is necessary to scrutinize some assumptions that are tacitly made in standard courses. For instance, some concepts such as thermodynamic equilibrium are not well defined. Also, the selection of macroscopic variables that determine the state of the system is treated only empirically, depending on the processes we are interested in.
We first give answers to the entropy production within a phenomenological approach, the thermodynamics of irreversible processes. We introduce relevant degrees of freedom that characterize the nonequilibrium state of the system. This is based on the discrimination between slow and fast processes and the degrees of freedom are introduced correspondingly. Expressions for the entropy production can be given for transport processes as well as for relaxation processes. Obviously, we have situations where the entropy production is unambiguously described on a phenomenological level. This helps us to find a general approach to nonequilibrium evolution.
It is possible to formulate an approach where the result does not depend on the arbitrary subdivision into “macroscopic” (relevant) and “microscopic” (irrelevant) degrees of freedom, the Zubarev method of the nonequilibrium statistical operator. An infinitesimal modification of the von Neumann equation is considered. A source term is added that goes to zero after the thermodynamic limit. However, no rigorous mathematical proof is known as to whether the problem to derive macroscopic irreversible behavior on the basis of “microscopic” reversible equations of motion can be solved this way. Irrespective of this, we will use this approach later on as the most recent promising approach to describe different nonequilibrium situations.
We use the formalism known from standard courses; some of the relations given here will be detailed in the subsequent chapters.
1.2.1 Quasiequilibrium
In contrast to the microscopic description that solves certain models considered to reproduce a given system, thermodynamics is directly related to real systems and gives phenomenological relations that are well defined. It is based on three “laws”; however, it also contains some more assumptions that are challenging.
Thermodynamic equilibrium occurs when any system's macroscopic thermodynamic observables have ceased to change with time. It is a basic concept considering a closed system. If we wait for a sufficiently long time, no further changes will occur in the system once equilibrium is established. However, this cannot be considered as a definition of thermodynamic equilibrium. How long do we have to wait? What about fluctuations?
Irreversible processes can occur in the isolated system. Typical irreversible processes are the degradation of differences of temperature, concentration (diffusion), and flow (friction). As a result, the inhomogeneities will disappear, but some properties such as energy, total momentum, total angular momentum, particle number, and volume are conserved and remain unchanged at all times. These are natural variables of the isolated system. As an example, we can consider a gas of identical molecules studied within the kinetic theory of gases. The equilibrium state (densities of particle number and energy are constant in space, no flows) is assumed to be stable. Any deviation such as a density fluctuation will relax to its equilibrium value.17 Typical relaxation times of density gradients for a gas at room temperature and normal pressure are very short (collision frequency of the order 109/s).
However, this kinematic equilibrium must not include chemical equilibrium. For instance, oxyhydrogen gas is explosive, and only after an explosion the chemical equilibrium can be established. Nevertheless, oxyhydrogen has properties like other gases; we denote this as metastability. We have to wait for a very long time before full equilibrium is established because the reaction rates under normal conditions are very small.
Similar problems are familiar with respect to a glass that behaves like a system in thermal equilibrium, but the amorphous state is metastable and may crystallize after long time, on the order of hundreds of years (glass windows in churches from the Middle Ages). In principle, exact equilibrium will hardly ever be achieved.18
These examples show that there is no exact formulation of thermodynamic equilibrium because “sufficiently long time” is not well defined. The better approach is to consider any system as a nonequilibrium system. If there are slow processes, we can consider “quasiconserved” quantities that relax slowly compared to the kinematic degrees of freedom. With respect to the element distributions in the universe, this seems to be quite natural. Chemical reactions are sometimes slow so that the composition is frozen-in. In chemical kinetics, reaction rates are approximately calculated assuming a momentum distribution of the components that corresponds to thermal equilibrium.
Another case is local thermodynamic equilibrium where we have well-defined local densities of energy , mean velocity , and particle number that smoothly depend on position . As local densities of conserved quantities, they can change only due to currents. Under certain conditions, this may also be a slow process compared to molecular collisions that lead to the local thermodynamic equilibrium. Compared to the rapid formation of local thermodynamic equilibrium, the densities change with time only slowly. We discuss this in more detail in the thermodynamics of irreversible processes, section 1.2.3, and in connection with kinetic theory, chapter 4.
We come back to the nonequilibrium case where the composition is frozen-in. The Hamiltonian of the system is taken as
(1.54)
where are creation and annihilation operators, respectively, for particles, species (including spin) and momentum and . We consider only elastic collisions, , where the participating particles are not changed. Reaction processes described by the remaining terms of are considered as small perturbations that are neglected.19 In this approximation, we have conserved quantities, the total energy , and the total particle number that commute with .
We solve the von Neumann equation to calculate the statistical operator. In thermal equilibrium, the statistical operator does not depend on time, since in thermodynamic equilibrium all properties, including the probability distribution, will not change with time. Therefore, it holds that and, according to the von Neumann equation, . This means that is a function of the conserved quantities. With this trivial result, the von Neumann equation is fulfilled. To find a special solution, we have to specify the initial condition (as is well known for the first-order differential equations in ).
We need a new principle to determine the initial state. This is the maximum of information entropy. For an open system, we can prescribe the averages at time ,
(1.55)
and look for the maximum of the information entropy that is compatible with the consistency conditions (1.55). We find the normalized statistical operator (see Eq. (1.30) and section 1.2.2):
(1.56)
where the Lagrange parameters that are determined by the consistency conditions (1.55) have the meaning of temperature and chemical potential of species .
Statistical physics is well developed for thermodynamic equilibrium. Any property can be calculated, such as the equations of state that relate different thermodynamic variables. The new principle needed to formulate statistical physics is the principle of maximum information entropy. This allows us to determine probability distributions to find a realization of a thermodynamic macrostate, with given averages as constraints (Problem 1.15).
We come back to the important point: Are the particle numbers of the different components exactly conserved? If they are slowly changing with time due to reactions, the consistency conditions (1.55) also depend on time . Assuming quasiequilibrium, the distribution is also given by the grand canonical ensemble (1.56) that depends parametrically on because the Lagrange parameters are determined by the self-consistency conditions (1.55). The calculated values for , in particular for , will depend on . (They are not constants like or .) Consequently, the statistical operator for the grand canonical ensemble will also depend on via the actual values of the Lagrange parameters , that is, in a parametric way.
This phenomenological approach can be applied only in the limit where the particle numbers are nearly conserved. A consistent approach that considers the solution of the von Neumann equation will be given at the end of this chapter.
1.2.2 Statistical Thermodynamics with Relevant Observables
Thermodynamics introduces variables that describe the state of the system. We will denote this as the relevant observables that characterize the macroscopic state of the system. In equilibrium, we identify them as the conserved properties that are not changed by the internal dynamics of the system: energy , total momentum , total angular momentum , total charge , total number of particles of species , and so on. They will be changed by external influences such as fields or contact with reservoirs. In particular, the (reversible) exchange of energy with a bath is denoted as heat , the exchange of particles as chemical work , the exchange of volume as volume work , the exchange of momentum as mechanical work , and so on. Macroscopically, we introduce the concept of ensembles that are characterized by the different contacts with the surroundings – no contact: isolated, microcanonical ensemble; thermal (only exchange of energy, heat): diabatic, canonical ensemble; heat and particles: open, grand canonical ensemble; and so on.20
The first and second laws of thermodynamics relate the change of internal energy to the different contributions due to the coupling to external reservoirs. The changes of state variables are assumed to be slow, quasistatic, and reversible. Thus, irreversible contributions such as friction, turbulence, and so on are avoided, but the exact definition of “sufficiently slow” is missing (see the definition of equilibrium).
A more realistic, satisfactory approach is to allow time-dependent averages,
(1.57)
of a set of relevant observables that describe the macroscopic state of the system. We take a time instant and describe the state of the system with . We work out a theory that allows extending the set of relevant observables so that we are not forced to select a prescribed reduced number of relevant observables.21 We also aim for arbitrary time dependence. Here, we consider the limit of “slow” processes where the system remains near thermodynamic equilibrium. We will analyze the corrections due to time dependence later on.22
For a given set of mean values of the relevant observables , one can construct more than solely one partition function. In other words, the partition function is not uniquely determined by the given averages. We construct the so-called relevant statistical operator (or partition function) using the same arguments as in equilibrium statistical physics, in particular the principle of maximum of information entropy.
We find the generalized Gibbs distribution:
(1.61)
where the Lagrange multipliers (thermodynamic parameters) are determined by the self-consistency conditions:
(1.62)
is the Massieu–Planck function, needed as a Lagrange parameter for normalization purposes and playing the role of a thermodynamic potential (Problem 1.16). Using the properties of the now defined relevant statistical operator, we find the following for the entropy of the relevant distribution:
(1.63)
The definition of provides
(1.64)
and from Eq. (1.63) follows:
(1.65)
The symbol denotes the functional derivative. Next, we derive an equation of motion for the relevant entropy . Since is a functional of the relevant observables , the time derivative of yields
(1.66)
With the thermodynamic relation (Eq. (1.65)), we obtain
(1.67)
The Lagrange multipliers are determined by the consistency conditions (1.57). The relevant statistical operator is not the correct statistical operator for the nonequilibrium because it does not reproduce the averages of the irrelevant observables. Furthermore, it is not derived as a solution of the von Neumann equation. At the end of this chapter (section 1.2.5), we construct a nonequilibrium statistical operator23 that solves the von Neumann equation, using the relevant statistical operator .
1.2.3 Phenomenological Description of Irreversible Processes
We consider sufficiently slow processes. This means, we assume that the system is always near to quasiequilibrium (local equilibrium). This special situation is treated by the thermodynamics of irreversible processes.24
We consider a multicomponent system ( components). The equilibrium state is characterized by the mean values of conserved quantities , such as energy and particle number of species . These quantities are additive and can also be attributed to a (macroscopic) partial volume of the system. denotes the amount of in . We consider homogeneous systems (thermodynamic limit) where we can introduce densities. Instead of extensive quantities of the system that diverge in the thermodynamic limit, the densities remain finite, intensive quantities. We define densities (e.g., energy density and particle density ) as limiting values:
(1.68)
where we assume that the limit is performed such that the volume element remains macroscopic, that is, the densities only weakly depend on position .
Conservation laws are global. In a finite volume element , the globally conserved quantities like particle numbers can change by flow through the surface of the volume element . The variation of densities with position can induce currents that lead to an equalization of the differences. The current density is defined in such a way that an area is chosen that is perpendicular to the flow of the quantity under consideration (the normal is directed parallel to the flow of ).
We define
(1.69)
where is the flow velocity of the property (see Figure 1.1).
Figure 1.1 Definition of the current density of property C.
The current densities of the conserved quantities follow the local balance equations:
(1.70)
Usually the mass current densities of the component , mass , is defined with respect to the barycentric system, . Here, denotes the center of mass velocity, . Then, the sum of the mass current densities is zero, .
Also, for quantities that are not conserved, we can formulate local balance equations. We decompose the temporal change of the density of the observable in the volume into a part that arises from a flow into or from the outside, and into a part due to processes within the volume :
(1.71)
Using Gauss' law, we express the contribution of the flow through the surface of the volume by an integral over the volume (jB defined like Eq. (1.69)):
(1.72)
The change due to processes within can be expressed in terms of a source density (production density) ,
(1.73)
Because the interval of integration can be chosen arbitrarily in the balance equation (1.71) with (1.72) and (1.73), the integrands must coincide:
(1.74)
This form of the balance equation can also be considered as definition of the production density . These production densities vanish for conserved quantities (see Eq. (1.70)).
The central quantity of the thermodynamics of irreversible processes is the production density of entropy. The entropy production within a system is responsible for the irreversible change of the entropy. To formulate the balance equation for the entropy, an essential assumption is local thermodynamic equilibrium. This means that in each volume element , the equilibrium relations between the state variables, that is, the equations of state, hold. Thus, we avoid the difficulty of defining thermodynamic quantities such as temperature or entropy in nonequilibrium. However, quantities like temperature or chemical potential become functions of position, or , respectively.
In each volume element , the Gibbs fundamental relation holds:
(1.76)
Then, the entropy current density results as
(1.77)
where is the current density of heat. Using the general relation
(1.78)
between the local and total variation of densities with time as well as the balance equations for the particle densities and the density of internal energy , we obtain the balance equation of the entropy (Problem 1.18):
(1.79)
The production density of entropy
(1.80)
has the general form
(1.81)
because of the second law. The production density of entropy is proportional to the current densities . The relation (1.81) defines the generalized forces as the coefficients of the respective current densities . For example, according to Eq. (1.80), the generalized force with respect to the heat current density is the temperature gradient , and with respect to the particle current density , the generalized force is the gradient of the chemical potential .
In thermodynamic equilibrium, the current densities vanish. The generalized forces are also equal to zero in thermodynamic equilibrium, . In particular, the generalized forces defined in Eq. (1.80) are given by gradients that vanish in a homogeneous system where and are independent of the position . We make the phenomenological ansatz:25
(1.82)
the current densities linearly depend on the generalized forces. This realizes the condition for . The phenomenological coefficients
