Entropy is in Flux
The science of thermodynamics was put together in the Nineteenth Century to describe large systems in equilibrium. One part of thermodynamics defines entropy for equilibrium systems and demands an ever-increasing entropy for non-equilibrium ones. Since thermodynamics does not define entropy out of equilibrium, pure thermodynamics cannot follow the details of how this increase occurs. However, starting with the work of Ludwig Boltzmann in 1872, and continuing to the present day, various models of non-equilibrium behavior have been put together with the specific aim of generalizing the concept of entropy to non-equilibrium situations. This kind of entropy has been termed kinetic entropy to distinguish it from the thermodynamic variety. Knowledge of kinetic entropy started from Boltzmann’s insight about his equation for the time dependence of gaseous systems. In this paper, his result is stated as a definition of kinetic entropy in terms of a local equation for the entropy density. This definition is then applied to Landau’s theory of the Fermi liquid thereby giving the kinetic entropy within that theory.
The dynamics of many condensed matter system including Fermi liquids, low temperature superfluids, and ordinary metals lend themselves to the definition of kinetic entropy. In fact, entropy has been defined and used for a wide variety of situations in which a condensed matter system has been allowed to relax for a sufficient period so that the very most rapid fluctuations have been ironed out. One of the broadest applications of non-equilibrium analysis considers quantum degenerate systems using Martin-Schwinger Green’s functions as generalized of Wigner functions, and . This paper describes once again these how the quantum kinetic equations for these functions give locally defined conservation laws for mass momentum and energy. In local thermodynamic equilibrium, this kinetic theory enables a reasonable local definition of entropy density. However, when the system is outside of local equilibrium, this definition fails. It is speculated that quantum entanglement is the source of this failure.
- 1 "Taking Thermodynamics too Seriously"
2 Boltzmann: Three Kinds of Entropy
- 2.1 Thermodynamic entropy
- 2.2 Analysis of the Boltzmann Kinetic Equation (BKE).
- 2.3 Two interpretations of
- 2.4 Three kinds of entropy
3 Landau’s Kinetic Equation: Boltzmann Reapplied
- 3.1 Metal physics
- 3.2 Quasiparticle excitations
- 3.3 Quasiparticle scattering: The right hand side
- 3.4 Quasi-particle energy: The left hand side
- 3.5 Quasiparticle entropy
4 Seeking Quantum Kinetic Entropy
- 4.1 Introducing thermodynamic Green’s functions
- 4.2 The Maths
- 4.3 From Boltzmann back to Maxwell (again)
- 4.4 Variational replacement
- 4.5 Number Conservation
- 4.6 Momentum conservation
- 4.7 Entropy
1 "Taking Thermodynamics too Seriously"
Professional philosophers of physics like sharp definitions; theoretical physicists tend to be more loose. These facts should not be a surprise to anyone. Philosophers want to bring out in a precise and careful manner the actual assumptions and content that form the basis for physical theorizing and physical theories. In the practice of “normal science,” scientists are trying to push back the borders of their domains by extending the range of application of their theories. Thus, they would like some flexibility in their basic definitions so that they can change their content without too much intellectual pain.
These contrasting tendencies have long conflicted in making definitions of entropy. It is not accidental that the very first use of the word “entropy” in English, by George Tait, was prefaced by the statement that he was going to use use the word in the opposite sense from its German inventor, Rudolf Clasius.
This paper was occasioned by thinking about a work by the philosopher Craig Callender entitled “Taking Thermodynamics Too Seriously.” His basic point was that one should not apply thermodynamic concepts to all kinds of things, but one should stick to the original area of application of thermodynamics222My difference with Callender is precisely that he takes thermodynamics too seriously. Following the wording and emphasis of Callender I would say that “The problem is […] thinking that one is explaining the behavior of individual real systems by ” appealing to the laws of thermodynamics. Indeed, thermodynamics applies to nothing in this world but instead to the limiting case implied by the words “the thermodynamic limit”.. This area was carefully defined by J. Willard Gibbs in about 1875 in his treatise. Gibbs described the situation in which thermodynamics laws would hold, the thermodynamic limit, as one in which one would have a homogeneous system, tending to be infinite in extent, and having been left in a constant external environment for a time tending to be infinitely long. In this view, any effect of finite size, transient behavior, or observable non-uniformity in structure would push the object under study out of the realm of thermodynamics. The homogeneity of the system is important since entropy should be an extensive quantity, namely one which grows linearly with size as the system grows larger.
Sadi Carnot’s and Rudolf Clausius’ thermodynamic analysis provided the original source of this entropy concept. It is reasonable to call the entropy that comes from their ideas and Gibbs’, thermodynamic entropy. However, beginning with Boltzmann the concept of entropy has been extended far beyond its original boundaries, so that at present the word “entropy” appears in the consideration of black holes, small- dynamical systems, entanglement, and information theory. Indeed perhaps, some authors extend this concept beyond recognition. This paper traces looks at the simplest and most traditional extension, that of kinetic entropy.
1.1 An alternative definition of entropy
In my view as a theoretical physicist, the essential question is not one of precise and careful definition. Many such definitions are possible and plausible. Instead one should ask a question which has a answer, for example, Is it possible to define, calculate, and use an entropy for a broad class of non-equilibrium systems. For a working scientist, it is important that the definition be useful for gaining an understanding of the workings of the world, and also for constructing further scientific knowledge. Nonetheless let me try to give a careful definition of entropy alternative to that of Callender and the thermodynamicists. I start:
A dynamical system is a set of equations describing the state of a idealized system aiming to emulate the dynamics of matter.
The system has a “state function” which describes its state in the neighborhood at the space point, at time . This function may have many components and may depend upon many subsidiary variables. This state function obeys an autonomous differential equation. Here “autonomous” means that the form of the differential equations depends upon neither nor but only upon
To define kinetic entropy, we imagine three functions of the state function: two scalar, the entropy density and a collision term RHT, and a spatial vector , all of which have no explicit space or time dependence. They vary in space and time because they each depend upon the values of in the neighborhood of the space point, , and they are respectable, reasonably smooth functions of .
They are related by the equation
where the right hand term (RHT) is the descendant of Boltzmann’s collision term. It is always positive333For appropriate functional forms of the distribution function, positive, finite, and sufficiently smooth. except when for all and some finite interval of times get a very special, time-independent form, called an equilibrium distribution. One can call the entropy thus defined kinetic entropy to distinguish it from thermodynamic entropy.
Under these circumstances, the is an entropy which describes the non-equilibrium state of the system.
Using this approach, Boltzmann defined entropy for a dilute gas using Eq.(1-1)(See Sec.(2).); while Landau set up a quasiparticle dynamic for some low temperature fermion systems that enable the definition of entropy (See Sec.(3).). On the other hand, we do not know whether field theoretic Green’s function methods[10, 11, 12] enable the definition of entropy in non-equlibrium situations.
If on the other hand, if it is impossible to construct an equation of the form of Eq.(1-1), but the best one can do with smooth and local functions is perhaps something like
then the concept of kinetic entropy is undefined away from the regions of the -space in which all the summation-terms are zero.
It should be noted that the constraints of smoothness and locality are crucial. It is likely that, dropping these constraints, one could define an entropy concept in which there was a global, increasing, entropy that might form a fractal web through the space entire system. Such an entropy might apply to entangled quantum systems, but would be vastly different from Boltzmann’s kinetic entropy.444In a chaotic classical system we have a large number of constants of the motion which have level sets in the form of a complex web through the entire coordinate space. When we talk about “conserved quantities,” we do not usually include these very complex objects. Perhaps similarly the existence of entanglement entropy might preclude the existence of Boltzmann-like kinetic entropy.
1.2 Outline of paper
The next chapter, Sec.(2), goes back to Boltzmann who showed how rarified gases could be described with the use of three conceptually different kinds of entropy. They are the thermodynamic entropy à la Clausius, the kinetic entropy that describes the gas’ relaxation to thermodynamic equilibrium, and a statistical entropy that is expected to be the logarithm of the number of micro-states subsumed by a single macroscopic configuration. All three entropies can be defined to have the same numerical value in thermodynamic equilibrium. However, Boltzmann’s three-fold extension was obtained at the cost of an assumption that only applies in a very special limit: one in which the interaction among the gas molecules be so weak as to provide no direct effect upon the gas-state thermodynamics.
In the Landau theory of a Fermi liquid[8, 13, 14], described in Sec.(3), all interactions are exactly included but one assumes that the system has the smoothness and homogeneity of a fluid. The validity of the Landau theory also relies upon low temperatures to force the system into a dynamics based upon excitations that can be described as quasiparticles[16, 17]. In this case also, three kinds of entropy can be calculated and agree with one another in appropriate limits. Once again, as in Gibbs description of thermodynamics, one must make assumptions about the system being sufficiently large and having had sufficient time-left-alone to relax the most complex correlations.
Field theory[15, 10, 12] provides a more general quantum theory that can be shaped into a quantum version of the Boltzmann kinetic equation. This generalized Boltzmann equation includes both frequencies and momenta in its description of excitations. I expected this approach to provide a more general derivation of the three three kinds of entropy. So far, I have failed to achieve this. Instead, I have much weaker statement derived from this point of view, namely the derivation of kinetic entropy, Eq.(1-1), seems to hold only in the limit of local thermodynamic equilibrium. However, the approach in Sec.(4) seems to give in general Eq.(1.1) rather than Eq.(1-1). Therefore the first two approaches (Boltzmann and Landau) are fully consistent with the definition of kinetic entropy but the last may well not be so.
2 Boltzmann: Three Kinds of Entropy
2.1 Thermodynamic entropy
The concept of entropy was put together in the nineteenth century by such giants as Carnot, Clausius, Boltzmann, and Gibbs. They constructed a beautiful and logically complete theory. Their work reached one kind of culmination with the 1875 and 1878 publication of J. Willard Gibbs, which described the entire theory of thermodynamics and of the “thermodynamic limit”. This is the situation which would arise in most macroscopic systems if left alone for long enough and thereby allowed to relax to a steady situation. Each such isolated system has a property called entropy. The entropy of a totality is the sum of the entropies of its individual isolated parts. If such a system is driven out of equilibrium, for example by bringing together two of its parts initially at different temperatures, the system will return to equilibrium, but in the process one observes that its entropy has increased. It is, of course, natural to ask whether entropy can be defined during the period in which the system is out of equilibrium. This paper discusses partial answers to this question.
2.1.1 Maxwell and the basics of flow and collision
When two molecules come within a certain distance of one another a mutual action occurs between them which may be compared to the collision between two billiard balls. Each molecule has its course changed and starts on a new path. James Clark Maxwell. Theory of Heat[page 302, original text 1871]
James Clark Maxwell was one of the founders of the kinetic theory of gases. His work arose before an atomic theory was generally accepted, but nonetheless he (and his near- contemporary Ludwig Boltzmann) saw a gas as a multitude of very small particles in rapid motion. These particles each would move in a straight lines, each exhibiting constant momentum, , and velocity, except that
they would gradually be deflected by slowly varying forces, like those from gravity and electromagnetic fields
and also, they would very suddenly change their direction of motion as they bumped into one another555This breakup of the forces in the system into two parts, analyzed differently, follows the classic work of van der Waals..
Maxwell was particularly interested in the transport of energy and momentum through the system, since this transport could be studied in the laboratory. For this purpose he constructed what he called transport equations, describing the motion of observable quantities through his gases. These observables were functions of the momenta, , of the individual gas particles, with being an index defining a particular particle of the gas. The observable might then be the momentum itself, , its position, , the particle’s kinetic energy, , or indeed any function of the particle’s position and momentum. The transport equations then described how the observable quantities might change in light of these processes just mentioned. When he would focus upon a given momentum-value, , Maxwell could enumerate the number of collisions per unit of time which would have the effect of knocking particles out of that particular momentum-value and equally, he could keep track of the number of particles entering that value in virtue of collisions. When all this was done, Maxwell could estimate, or in a particularly favorable case, calculate the rate of change of his observables in time. Thus, he could describe both the dynamical processes involving the transport of energy, particles, or momentum through the gas. In addition, by looking at situations in which the state of the gas was unchanging, he could also define the nature of the equilibrium of the system, via the famous Maxwell or Maxwell-Boltzmann distribution.
Boltzmann took Maxwell’s elegant but limited calculations and converted them into a generally formalism for the evaluation of gas properties. He changed the focus of the calculation from the consideration of particular observable properties of the gas to a calculation of the rate of change of an average number of particles in a given region with a given momentum, . Maxwell’s and Boltzmann’s original definition666Boltzmann’s original  definition actually used the kinetic energy as the independent variable in his analysis[page 89]. This is an not an elegant choice compared to , and has not been used extensively by later authors. of this quantity[19, 7] is that
is the number of particles we will find at time within the small volume of ordinary space and the small volume of a space describing the possible momenta of the particles. However, since many of their calculations involved averaging processes, this definition did not stand up to closer analysis. Eventually, Boltzmann changed his interpretation to one involving probabilities. Gibbs completed the analysis in his work on statistical mechanics . These authors interpreted probabilistic calculations, including ones involving , by considering the average behavior over many possible similarly prepared systems. Such a collection of systems eventually came to be called an ensemble, and this kind of calculation statistical mechanics.
Using the ensemble definition of , we can say that the total, averaged, amount of observable per unit volume in the region around at time could then be defined as
Here indicates an average over the ensemble. (Note that we integrate over possible momenta, , , while holding the space and time coordinates constant.) Boltzmann’s calculational strategy was then to first write an equation to calculate the distribution function, , rather than the individual particle positions and momentum, and then use to calculate the average of observables.
In either view, deterministic or statistical, Boltzmann’s -function exists in six dimensions plus time. ( Indeed, Boltzmann was a quite brave theorist. No complexity fazed him. )
Even the original incorrect, deterministic, definition of was an important advance. It permitted Boltzmann to write down his celebrated “Boltzmann kinetic equation, “BKE,” from which he was able to deduce the main properties of gases and to invent statistical mechanics. I here write down this equation in modern language, so that I may describe its contents. The equation is
The BKE describes fully the behavior of low density gases composed of structureless particles interacting via short ranged forces. It enabled Boltzmann to describe the main dynamical properties of these gases and to construct the basis for a tremendous amount of further work in the dynamics of condensed matter systems. Further, equations with a similar form have been used to describe the kinetic properties of a wide variety of different condensed matter systems up to the present day.
Starting from the left-hand side of Eq.(2-3), we see the time derivative of the distribution function . The next term describes the gradual changes in momentum and position of the particles described by . This term reflects the formulation of mechanics described by William Rowen Hamilton in 1833. Here, is the energy of a single particle as it depends upon particle momentum. It may also include some space- and time- dependent potentials, as for example electromagnetic potentials. In that case, it is possible for the energy, , to dependent upon space, and time, . The bracket is called a Poisson bracket and is defined by
In the BKE of Eq.(2-3), is the one-particle Hamiltonian and is , and this Poisson bracket describe how gradually changes with time. The change occurs because each particle’s position variable, changes as it moves with the Hamiltonian velocity777 The reader might worry that the signs coming from the Poisson bracket of BKE are the opposite the signs in Eq.(2-5). That is because the variables and in are not directly the position and momentum of any particle. Instead they are parameters defining which particle momentarily appears with these phase space coordinates at the instant T. The definition is
|while its momentum is varying in virtue of the force|
The right hand side of the Boltzmann equation describes collision processes. In the first term, proportional to , the quantity describes the number of scatterings of particles with momentum per unit volume and unit time in the neighborhood of the space-time point . The minus sign, of course, indicates that this process decreases the number of particles with momentum . The remaining term, involving describes the opposite process, that is the scattering of particles that then go into the momentum state, . Thus, measures the number of particles going through this process per unit volume and unit time. Both scattering terms are complicated integrals involving four different momenta, two before the collision and and two after. These terms make the BKE into a non-linear integro-differential equation, quadratic in .
These scattering terms are most effectively described by setting down the total rate of scattering of all the particles in the system. (See [pages 88-93].) The rate at which particles with momentum and scatter off one another and end up respectively with momenta and can be written as a functional of Boltzmann’s distribution function, . The total scattering rate is proportional to
|In an appropriate limit of a low density gas with short ranged interactions the collision rate is exactly proportional to this expression, with an appropriate . (This limit is described as the Grad Limit.) The factor of one half is inserted for later notational convenience.|
This process is depicted in Figure(1). The function describes the exact scattering rate and is derived from a solution of the equations of motion obeyed by the colliding particles. We shall not specify it in detail here, but only focus upon the symmetries which must be obeyed by this quantity for the Boltzmann equation to make any sense. In Boltzmann’s work was specified to describe the result of the classical-mechanical analysis of the two-particle scattering
This can then we used to find the collision terms in the Boltzmann equation. When is properly normalized, we read off the scattering rate from as
This same logic gives us an expression for the rate of scattering into the momentum :
It will be important for us that the collisions include the conservation of particle number, momentum and energy. The number conservation is built into the structure in its demand for having precisely two particle both before and after the collisions. We further demand that be zero unless the conservation rules are satisfied. This will occur when is proportional to Dirac delta functions in the form
where depends upon combinations of the momentum variables not fixed by the delta functions.
2.2 Analysis of the Boltzmann Kinetic Equation (BKE).
2.2.1 Symmetries of the BKE
When we substitute the results of Eq.(2-6b) and Eq.(2.1.2) into the expression of Eq.(2-3) we have a formulation of the equation called the Boltzmann’s kinetic equation, which will then form the basis of the rest of this paper. However our formulation is slightly different from the BKE originally used in 1884. The original work had a much more explicit form for the collision function, , based upon solving equations of classical mechanics for two particles and a central force. One advantage of our more generic formulation is that we do not have to look into the details of the scattering calculation. A subsidiary advantage of not specifying explicitly is that the BKE thus written applies very broadly. In fact, it equally applies to quantum gases in the low density limit, with the previously stated requirement is that the intermolecular forces be short ranged. In the quantum case, one uses the differential cross-section to specify the scattering rate.
Since the structure of , and not its details, matters to us here, let us outline the needed symmetries . These include
Interchange symmetry. For simplicity we assume that the scattering particles have identical properties so that
Detailed Balance. Another important symmetry is one in which the primed particles come together, collide, and then the unprimed ones come off. If scattering rates for this kind of event (see Figure(1)) are the same as the rate for the corresponding direct process
the system is said to have a detailed balance symmetry.
Positivity. is always positive or zero.
Galilean symmetry. The scattering rate will be the same for an observer at rest as for one in motion with velocity . If the mass of both scattering particles is then
Rotational symmetry. If all the momentum vectors undergo the identical rotation, remains unchanged.
Time reversal symmetry. This holds in the absence of magnetic fields. It is represented by . Under this transformation, remains unchanged.
Parity, Flipping the sign of an odd number of components of all the momenta leaves unchanged.
2.2.2 Boltzmann’s conserved observables
The next step is to write down equations for the time dependence of observables, and thereby trace the route from Boltzmann’s kinetic equation back to Maxwell’s transport equations. Multiply the kinetic equation, Eq.(2-3), by and integrate over to find
Note that, for simplicity, we have taken the observable to have no explicit time dependence. The observable, then, only dependence upon time arises from particles’ motion in phase space.
There are three terms in Eq.(2.2.2). The first is simply the time rate of change of the observable. The second,
comes from the time-dependence of in virtue of the one-body Hamiltonian . The divergence term in Eq.(2-9a) describes the rate of change produced by the flow of the observable, which then produces a current
The bracket term on the right of Eq.(2-9a) is the intrinsic change in the observable produced by the time derivative of the positions and momenta of particles in the neighborhood of . We describe this term as the negative of a generalized force, . Then the entire kinetic equation is the statement
with the right hand terms, RHT, being the collision terms,
2.2.3 Collision analysis–variational method
We follow Boltzmann and look at the observables most significant for understanding the long-time behavior of the system. These special observables are the conserved quantities, those which are time-independent in an isolated system. They are the ones which are not changed by collisions terms, so that, for them, RHT
We could analyze the collision terms quite directly using the symmetries we have just listed. Instead, we employ a method based upon variational analysis. This method is somewhat too ponderous to be the most elegant method of understanding the BKE. However, the very same method will come into it own in the analysis of quantum problems, so we introduce it here.
We consider the total scattering rate as a functional of both the density of scatterers, which will be indicated here as and the density of final states, denoted by . The total scattering rate is then proportional to
|To get the rate of scattering of a beam of particles with momentum we calculate a derivative as|
|In precise analogy, to calculate the scattering rate of particles into the beam, we use|
This formulation is particularly useful in evaluating the combination
|which has the effect producing a sum of terms in which each term has a replaced by or in which a replaced by , that is|
(We have used the symmetry of Eq.(2-7a) to form this result. This symmetry is implicit in our use of the variational derivative to form and .) We describe the analysis in Eqs.(2-11) as a variational replacement. We shall see similar replacements in our fully quantum analysis of Sec.(4.4)
Specifically, the combination used in calculating the effects of our observables upon the right hand side of the Boltzmann equation is of this form with and . As a result we find that the effect of the observable upon the right hand side of the Boltzmann equation is
We can now simply read off conserved quantities by noting the delta functions (see Eq.(2.1.2) ) in the collision terms.
The conservation laws are defined by the values of that make the integrated right hand terms, as defined by Eq.(2-12) equal to zero. These include the momentum (), angular momentum (), energy (), and particle number, ().
It is particularly crucial that any approximate kinetic theory include variables that obey these conservation laws, since the conservation laws drive the low frequency, long wavelength behavior of the system. For macroscopic observers, like ourselves, this slow variation dominates the immediately accessible behavior of the many-body system.
A very similar approach will work for the derivation of an equation for the entropy density. Boltzmann multiplied the Boltzmann kinetic equation, Eq.(2-3), by an observable that generates the thermodynamic entropy density,
and integrated over all momenta. The resulting equation has almost but not quite the form of a conservation law. It is
This equation describes a continually growing (or constant) entropy since the right hand side is greater than or equal to zero, and the left hand side describes how the entropy, with density ,
is moved from place to place by the current .
Since only the collisions increase entropy, the generalized force associated with the entropy density is zero.
The crucial analysis is that of the collision term. We once again use the analysis of Eq.(2-11) to get an expression for what we hope to be the entropy production rate in the form
Following in the footsteps of Boltzmann, we should make two demands of our putative expression for the local entropy production rate: It should be zero for systems in equilibrium and positive for systems out of equilibrium. The first requirement is automatically satisfied since the equilibrium (Maxwell-Boltzmann) condition upon is that be proportional to a sum of the conserved quantities
so that the term in square brackets in Eq.(2-14d) exactly vanishes in virtue of the delta functions in . When has its equilibrium form, we say that the system is in local equilibrium. Local because our derivation permits the inverse temperature, , the average particle velocity , and the chemical potential, to vary in space and time.
To complete his argument, Boltzmann found a sufficient, but not necessary condition for this non-negativity, specifically that obey the detailed balance condition of Eq.(2-7b), so that the collision process is equally likely forward as backward in time. In the case considered here, this detailed balance condition is a consequence of time reversal invariance together with the symmetry produced by reversing the direction of all the momentum vectors. If the balance condition is satisfied, the expression in the collision term of Eq.(2-14d), not including the last line, is anti-symmetric under the interchange of primed and unprimed variables. One can make the entire expression symmetric, without changing its values by making the replacement
After this replacement, there are two expressions in Eq.(2-14d) (the ones in square brackets), each functions of the ’s, both of which have a sign that depends upon whether is greater than or less then . Equality occurs only when these are equal, and this equality in turn happens only at local equilibrium. Therefore this right hand side is positive in all situations except when there is local equilibrium.
Boltzmann’s result is now often called the -theorem because later authors described the result in terms of , the negative of the entropy.
2.3 Two interpretations of
We have already noted in writing Eq.(2-1) that in the course of time Boltzmann stopped saying that was the actual number of particles in a given region of phase space, and instead came to interpret as an average density of particles in different regions of phase space.
The difference between defining a number of particles, and defining a probability is crucial to understanding the true meaning of the Boltzmann equation. It is the difference between a deterministic and a probabilistic definition of Boltzmann’s calculation. After the original publication of the equation, Boltzmann’s contemporaries, especially[pages 97-102] Loschmidt and Poincaré, pointed out that the Boltzmann equation could not be an exact representation of any situation in classical mechanics. Classical mechanics, they said, was reversible. If you reversed the direction of all the velocities in the system, the previous motion unfolded once more, but run backward! However, the Boltzmann equation exhibited no such behavior. The left hand side of Eq.(2-3) changes sign with this reversal; the right hand side remains unchanged. Clearly this equation’s behavior cannot reflect an exact property of any single configuration of classical particles. However, as Boltzmann realized and Howard Grad eventually showed, the equation could be describing a probability distribution and an average over similar systems.
2.4 Three kinds of entropy
So far we have followed Boltzmann’s development of a theory of kinetic entropy. This entropy is defined as an extensive quantity (i.e. a sum over locally defined quantities) with the property that it is continually increases until thermodynamic equilibrium is reached, whereupon it attains a constant value. It is distinguished from the thermodynamic entropy by being defined in out-of-equilibrium situations. By adding appropriate amounts of the other extensive conserved quantities to the kinetic entropy one can make the two entropies identical in equilibrium situations.
Boltzmann also found one more version of the entropy. In statistical mechanics, one averages over ensembles. The average has a weight that depends on all the variables in the system and their correlations. To calculate such an average, one uses a probability density in classical calculations or a density matrix in quantum calculations, both denoted as , and a sum over configurations called a trace. Thus the average of an observable is
In 1877, Boltzmann noticed that he could define a statistical version of the entropy that would agree with both of the other definitions for systems in equilibrium. This definition gives the entropy as minus the average logarithm of the probability density or density matrix
Here is the Boltzmann constant, set equal to unity in our other formulas. The statistical entropy of Eq.(2-16) can be considered to be yet a third kind of entropy.
Each of these different kinds of entropy has its own sharply defined range for validity: thermodynamic entropy, for large systems in equilibrium, kinetic entropy for low density gases, that is the Grad limit, statistical entropy, within the range of validity of statistical mechanics. However each one of these can be extended beyond this narrow range. Both the original ranges and also the extensions overlap. In fact, the ranges are continually being extended by scientific workers so that the concept of entropy is a growing rather than a static one.
3 Landau’s Kinetic Equation: Boltzmann Reapplied
The behavior of excitations in low temperature systems can be described in terms of quasiparticle kinetic equations that are closely analogous to the BKE. But unlike the BKE, which was mostly invented all at once, the understanding of quasiparticle dynamics was built up step by step over the decades between 1920 and 1960. This chapter traces a little of this development and then concludes with a discussion of entropy within the most transparent of these quasiparticle theories, the Landau theory of the Fermi liquid and its attendant Landau Kinetic Equation (LKE).
3.1 Metal physics
For decades following Boltzmann’s studies, the BKE seemed to describe a low density gas, but little else. The requirement that the kinetic equation describe accurately all the collisions that occur in the system could not easily be applied to any of the solids and liquids that surround humankind since al of them clearly involve the simultaneous interaction of many molecules. However, the charged excitations moving around in metals did not appear to behave totally differently from gases. Paul Drude’s[29, 30] work at the beginning of the Twentieth Century treated the electrical properties of metals in terms that were essentially similar to the kinetic theory ideas of Maxwell and Boltzmann. The major difference was that the particles in motion in metals were charged and therefore could respond to electromagnetic forces. Looking forward from Drude’s day, we can see how these forces might be included by inserting electromagnetic potentials in the left hand side of the Boltzmann equation. These forces would act continually upon the particles. The free motion of these particles would be interrupted by occasional collisions so that, as in a gas, the particles would diffuse through the material. To understand collisions, we might look to modifications of the right hand side of the Boltzmann equation.
Given the fact that a metal contains a dense periodic lattice of charged ions, each ion capable of strong scattering of electrons, the fact that the scattering would only happen occasionally was, after a time, recognized as quite unexpected.
One part of the puzzle was solved by Felix Bloch in 1927[pages 106-113]. He found that in quantum theory a particle moving in a periodic potential would indeed behave as a free particle. As in free space, these particles could have stationary states, each with a well defined energy. Each state would be labeled by a quantity, , that behaved much like a momentum. In this situation, the energy-momentum relation would no longer have the free space form, , but instead would form a periodic structure with the same periodicities as the ionic lattice of the metal. So one of the important activities in the physics of the middle of the Twentieth Century was to investigate, understand, and predict the form of this energy function, .
The usual metals support very strong Coulomb forces, that can produce bound states with a very large range of electronic energy eigenvalues. In the usual situation, the temperature is quite small in comparison to this eigenvalue-range. Early on, it was not clear how the electrons would arrange themselves in this set of configurtions. An entirely new insight was put together by many founders of quantum theory including Dirac, Pauli, and Summerfeld [pages 94-97]. They developed the view that electrons were particles that obeyed Fermi-Dirac statistics and that for such particles, it is impossible to have more than one particle per quantum state. So the states tended to fill up, starting from the lowest levels, which were completely filled, than a narrow band of partially filled intermediate levels, and then with the highest levels completely empty. This situation is not much disturbed by any perturbation. Hence these low-lying state are inert. The higher energy state are too high to have an appreciable occupation. They too remain inert. It is only the states within a narrow band of energies around a special energy, called the Fermi energy that are active. This band of energies is then the object of study in the electronic part of metal physics. As in low-density gases, one may describe the fermions in this situation by describing the density of particles labeled by momentum in the neighborhood of the spatial point . One might as well use the same notation as for a classical gas and describe this density by .
The major experimental tool for the investigation of the electronic structure involved the placement of the metal in electric and magnetic fields. The observation of the the resulting motion of the charged carriers could then give considerable insight into the properties of the electronic excitations. The experimental electric and magnetic fields were almost always slowly varying in space on spatial scales comparable to the distance between ions. They were also slowly varying over the characteristic times necessary for the quasiparticles to traverse the distance between ions. Under these circumstances, at low temperatures, the quantum theory for the excitation motion is is surprisingly simple: The carriers behave like classical free particles moving under the control of a Hamiltonian. The Hamiltonian itself is the as it is modified by the space- and time- dependent electromagnetic fields in the metal. These fields are produced by both the ions and electrons within the materials and also by slowly varying electromagnetic fields external to the material. These fields, or rather the potentials that produce them are reflected in changes in the quasi-particle energy . In this energy, the momentum variable is enhanced by the addition of electronic charge times the vector potential while the Hamiltonian itself has an additional term of charge times the scalar potential.
3.2 Quasiparticle excitations
The charge carriers in a metal are correctly described by this quantum analog of Boltzmann’s distribution function, . In the absence of collisions, once the effective Hamiltonian, , has been constructed, this particles obey Hamiltonian equations of motion in the form as given by Poisson brackets with . Thus the entire analysis that gives the left-hand side of Boltzmann’s kinetic equation, Eq.(2-3), applies equally to the dilute gas and to metals. Empirically, it is noticed that the charge carriers scatter rather seldom in pure metals at low temperatures. So the collision term is not of primary importance in determining the motion of the electrons. Excitations that move under the influence of an effective Hamiltonian like and which scatter rather infrequently are termed quasiparticles.888The actual word, quasiparticle, only comes into physics much after the initial work of Bloch. I have found the word in a 1947 paper of N. N. Bogoliubov on the behavior of dilute gases composed of Bose particles. Since quasiparticles provide a correct description of the low temperature behavior of many materials, quasiparticle analysis permeates condensed matter physics.
3.3 Quasiparticle scattering: The right hand side
Through the 1920s and beyond, scattering rates like the one Maxwell and Boltzmann used were employed to understand the scattering of particles, quasiparticles, and other excitations. The relevant quantum mechanical formula came under the name of the “Fermi’s golden rule.” It said that whenever the potential causing the scattering was weak the scattering rate could be computed999Much of the ideas for this comes from the work of Dirac. by
Here is the matrix element between initial and final states of the potential causing the scattering. Since the squared matrix element typically contains a delta function for momentum conservation we can recognize this golden rule expression as one possible source of the we used in the Boltzmann kinetic equation.
Since the early days of quantum mechanics, this approach has been used to understand the behavior of a continually expanding range of systems, including those of condensed matter. As this approach was developed, it was recognized that the scattering of the quasiparticles would end up as being quite important. The scattering might be infrequent, but they were necessary to bring the system into thermodynamic equilibrium.
In dilute ordinary gases, Boltzmann’s basic scattering approach could be carried forward to work within quantum theory. The scattering rate would be calculated by quantum theory, using a formulation like Eq.(3-2) with differential cross-sections replacing the squared matrix element. Thus, the basic structure of the kinetic calculation remained the same as the one introduced by Boltzmann.
3.3.1 Degenerate scattering rate
The most important addition that quantum theory brought to BKE type calculations came with the recognition that scattering processes would be modified to include “quantum statistics”. Quantum mechanical particles come in two kinds: fermions, which do not admit of more than one particle existing in a single quantum mode and bosons, in which the occupation of modes by multiple particles (or particle-like excitations) is enhanced. The very complex behavior which can be produced by these statistics nonetheless is reflected by very simple changes in the collision terms used in Boltzmann-like analysis. In a translationally invariant system, one can, as before say that the modes of the system can be labeled by momentum vectors, and that the number of modes available per unit volume in a volume in momentum space is , where is the spatial dimension and is Planck’s constant. We shall forget about the denominator, which can be absorbed into the normalization of the scattering rates and . So once more, we shall use the momentum vector to describe the scattering processes. In our further description we shall use, as before, to describe the density of particles, but now our normalization will make the average number of particles in each of the quantum modes in the neighborhood of the momentum . For Fermi-Dirac particles the density of available states will be reduced by the factor , to reflect the fact that multiple occupation is impossible. Bose-Einstein[33, 35, 36] particles have their scattering into a given range of modes enhanced by a corresponding factor of . To cover both cases, and to include nondegenerate situations, we shall take the scattering rate to be modified by the factor
with respectively for bosons, nondegenerate particles, and fermions. The so defined is the density of states for the excitations involved. In the case of fermions, it can also be described as the density of holes. For bosons this factor describes the enhancement of scattering which, for radiation, is described by the words stimulated emission. For symmetry, sometimes when we describe the density of excitations, , we shall write it as .
3.3.2 Variational approach
Classical formulas for scattering rates were defined in Sec.(2.2.3). A direct extension of our variational derivative description of BKE scattering as given in Eq.(2-6) is to define a to be a functional of both the density of quasi-particles and also the density of states for these quasiparticles, . In this writing we have
|and then get scattering rates via formulas analogous to Eq.(2-6b) and Eq.(2.1.2)|
|as well as|
|These results hold in second order perturbation theory with a two-body potential|
|Then will have the form|
Note that this result is automatically even (or odd) under the interchange of and for bosons (fermions).
3.3.3 Are quasi-particles well-defined?
Excitations in a translationally invariant non-interacting system may be classified by their momentum . With, for example, periodic boundary conditions this momentum is quantized so that one can keep track of individual single-particle states. Bloch showed that this is true also if the system has a static periodic potential. The idea of quasiparticles is based upon the view that as we gradually turn on interactions, these single-particle states will deform, but they will not change in any essential way. We know that this gentle deformation does not always hold. For example, if we break the periodicity by adding static impurities to a metal, it can contain localized single-particle states, with properties entirely different from the non-interacting excitations. Nonetheless, in the absence of evidence to the contrary, physicists usually assume that a translationally invariant material can be described by excitations with a well-defined momentum.
It turns out that at moderately low temperatures, this quasiparticle description is roughly true for a wide variety of materials. Its validity can be checked in part by calculating or estimating the inverse lifetime of the excitation using the scattering formulas of Eqs.(3-4). If this width times is much smaller than the range over which one has significant variations in the quasiparticle energy, then the quasiparticle point of view is likely to work. For example in a Fermi liquid (in particular He) the width of the active band of excitations is proportional to temperature, but the scattering broadens each excitation by only an amount proportional to temperature squared. Therefore, at lower temperatures, a quasiparticle theory is likely to be valid.
At the very lowest temperatures however all kinds of new dynamical phenomena are possible, so present theory has nothing definitive to say.
3.4 Quasi-particle energy: The left hand side
In 1956 Landau extended the ideas we are discussing by introducing a theory of the Fermi liquid, in part for application to the behavior of He at low temperatures[8, 13, 14]. The first of these papers is the one that defines the Landau Kinetic Equation and will be the paper relevant for our present discussions. The remarkable and new element of the LKE was the determination of the quasi-particle energy from the thermodynamic free energy. According to Landau, who offered us the equations to describe the Fermi liquid but did not say how they might be justified, the free energy, , is a functional of the distribution function such that the quasiparticle energy may be calculated as
Eq.(3.4) is the Landau Kinetic Equation, LKE.
The predictions from Landau’s theory and particularly the LKE were soon bourn out by the experimental data.
3.4.1 “Golden Rule” for correlation energy
Landau’s free energy includes the effects of correlations produced by the interactions among the particles. In parallel to the construction of a golden rule for interparticle scattering, there is an analogous and equally useful golden rule for correlation energy. Once again this result applies most directly to the case of relatively weak interactions among particles, but can be extended to apply, at least in a qualitative fashion, much more generally. In the golden rule calculation, which is essentially a result of second order perturbation theory, the correlation energy is of the form
The only difference from the generator of the collision terms as seen in Eq.(3-4a) is that Eq.(3-7) contains an energy denominator in place of Eq.(3-4a)’s energy delta function. The variational derivative of the expression of Eq.(3-7) with respect to gives the correlation contribution to the single-particle energy .
3.5 Quasiparticle entropy
The calculation of the time-dependence of the entropy density from Landau’s version of the kinetic equation, Eq.(3.4) is most simple and instructive. Although Landau makes use of the equation for the entropy derived below, for some reason he assumes its truth a priori and does not derive it from his kinetic equation. Let us imagine that we wish to calculate the time dependence of any function of the distribution function say , which has a derivative101010Here “S” is the standard symbol for entropy while “M” stand for multiplier.
|The equation describes what we now term as an entropy density|
|and an entropy current density|
So far we need not have specified the functions and . Eq.(3-9a) is a correct statement for any value of the function .The defining statement for the entropy is that the collision term for the observable, here described as the right hand term RHT, must always be greater than zero, except in local equilibrium where it will be zero. There is a choice that will meet the requirement that the RHT be non-negative in this fashion. That choice, unique up to an additive constant, is
|which then implies|
This choice will make the collision term have the form
which will then force this term to be non-negative whenever obeys the detailed balance symmetry of Eq.(2-7b). (See the discussion after Eq.(2-14d).) The only possibility of a zero is when is proportional to an exponential with exponent being a sum of local coefficient times the conserved quantities and .
As has long been known, the quasiparticle result binds together the different definitions of entropy. The calculation is a kinetic one, having the property that it holds into situations well out of equilibrium. It appears to remain true in all situations in which the quasiparticles are well defined. Of course the quantity defined by Eq.(3-9b) and Eq.(3-10b) is also a thermodynamic entropy since it works for equilibrium situations. Further, quasiparticle theory agrees with with the various laws of thermodynamics and part of that agreement is that the spatial integral of the quasiparticle entropy density, as present in Eq.(3-9b) serves as the thermodynamic entropy within that thermodynamics. Finally, this entropy fully agrees with a count of the number of quasiparticle configurations, as presented for example in Schrödinger’s book on statistical physics. Everything fits together under one roof.
4 Seeking Quantum Kinetic Entropy
4.1 Introducing thermodynamic Green’s functions
4.1.1 To a one-body analysis
The general form of a quantum non-equilibrium analysis is very complex indeed. For spinless particles one must specify a density matrix that is a function of separate variables when our system exists in the usual three dimensional space. However, analysis suggests that simpler situations are possible. In the last chapter we saw a quasiparticle state is specified by , a function of only six variables. At full equilibrium at a given temperature a state-specification requires of order a half-dozen thermodynamic parameters.
One possible theory for describing both equilibrium and non-equilibrium situations uses Green’s functions, . These functions are expectation values of creation and annihilation operators. This original equilibrium Green’s function theory specified by Martin and Schwinger  is based upon a time boundary condition due to Kubo. The result is that the Green’s function, , is composed of two functions, and , that both have a time dependence in which they depend only upon only upon the time difference variable, , with the times sitting in an interval . The function, G(1,2), can be split into two functions, and , with times that can be analytically continued onto the real line. These functions can be defined by a formally exact diagrammatic analysis using a Dyson equation[37, 38] and a set of skeleton diagrams that depict the Green’s function and the two-body potential. This equilibrium analysis employs one- and two-body potentials that are both functions of two spatial variables in addition to the time difference.. Once these potentials are given, the thermodynamic state is defined by roughly a half dozen equilibrium parameters, including temperature and chemical potential. Possible behaviors include many thermodynamic phases as well as various kinds of band structures and localized states. Many of these situations would give a behavior very different from that described by the Fermi liquid theory.
This Martin-Schwinger theory may be extended to describe time dependent phenomena, via a generalization of these same Green’s functions. Followup theory (See [10, 11, 12].) suggests that if one starts at equilibrium at some initial time, in a system specified by one and two particle potentials, and , then for subsequent times the system can still be be fully described by a one particle Green’s functions. These functions are specified by an equation of motion and further by a perturbation expansion containing an infinite set of diagrams. This situation is once again described by two Green’s functions, now written as and , that depend upon both time variables, and . A perturbation theory analysis once more generates a formally exact description of a self-energy to define these Green’s functions. I believe that time dependent situations are potentially more complex than static ones, but I do not know the degree to which this non-equilibrium situation can display a richer complexity of different phases or behaviors than is shown by the equilibrium case.
We are looking for a Boltzmann-like analysis. An additional condition is required for this approach to to apply. The functions and are all functions of a pair of space-time variables . These can then be expressed in terms of sum and difference variables as
As in the usual analysis of Wigner functions, a quasi-classical situation arises whenever the variation in the sum variables is much slower than that in the corresponding difference variables. In physical terms, the spatial variation of the sum variable must be slow in comparison to typical microscopic distances, including force ranges and than de Broglie wave lengths. The time variation in the sum variable must also be sufficiently slow so that we can consider all the excited energy levels in the system to be part of continua, rather than discrete levels. In this quasi-classical situation, the Green’s functions obey an equation very analogous to Boltzmann’s, with the important difference that excitations are described by both a momentum, , and a frequency, . Since there is a close analogy between the behavior of these two variables we shall often group them together as an energy-momentum variable, .
4.1.2 To a generalized Boltzmann equation
An excellent summary of the subject of this section is provided by Joseph Maciejko. For a summary of historical development and of applications see  especially the articles by Baym and Martin.
As pointed out in ref.[page 21] to derive the basic equation of the quantum kinetic theory one begins by assuming that the two forms of the Dyson equation:
both give the same (correct) answer for when they are applied in the time interval .
To do real time analysis of these equations, one uses Wigner functions. These are time-ordered versions of with times on the real axis. They are expressed in sum and difference variables, as in Eq.(4-1), and then Fourier transformed with respect to the different variables. The resulting real functions are , which describes the density of excitations labeled by in the space-time neighborhood of , and , which gives the corresponding density of available states in these variables. For fermions, can also be considered the density of holes. A density of states, uncorrected by occupancy, is given by
|(Recall that is plus one, minus one or zero respectively for bosons, fermions and particles obeying classical statistics.) For bosons, is greater than , indicating the enhancement of scattering caused by state occupation. For fermions, is smaller than . Finally, we shall make use of the propagator|
with Pr indicating a principle value integral.
The Dyson self-energy has a formally similar description. We define and as the two real-time, time-ordered components of . We use subsidiary quantities
|and the “real” (rather than complex) self-energy|
Here, is defined as the rate of scattering of an excitation out of a configuration in the neighborhood of , while is, after being multiplied by the density of states, the corresponding scattering rate into that configuration. Whenever the two-body potential, , is a delta function in time, with space Fourier transform, , the Hartree-Foch contribution to the self-energy, denoted as , is independent of frequency and cannot be represented in the same fashion as the rest of the self-energy. It has the form [pages 17-26]
in equilibrium in a translationally invariant situation. We shall henceforth assume this frequency-independent form for the Hartree-Foch term. The independence will cause it to disappear from many of our results.
4.1.3 Equations of Motion
We can see these definitions manifested in the generalized kinetic equations, which are obtained by subtracting the two forms of the Dyson equation, Eq.(4-2) and then doing a Fourier transform in the difference variables. In a first order gradient expansion, the resulting equations then read[40, page 27]
with all quantities being functions of and with the generalized Poisson bracket defined as111111Note that the part of this bracket has the opposite sign from the one used in the bracket we employed for the BKE and LKE.
These solutions for and are consistent with the result of taking the difference between second form of the kinetic equations Eqs.(4-6) and times the first form, which then gives
Eqs.(4-6) can be interpreted as simple generalizations of Boltzmann’s kinetic equation. On the right we see scattering terms, respectively describing the rates of scattering out of and into a configuration described by . The only thing that is new is the inclusion of the energy label, in addition to the momentum label . On the left hand side the generalized Poission bracket describes the rate of change of caused by gradual changes in particle position, momentum, and energy. Here is the propagator for the non-interacting system. It may include electromagnetic vector and scalar potentials as well as a one body potential . The self-energy, , includes all modifications of single-particle energies from particle interactions.
Finally, the equation for , Eq.(4-6a), contains the term . This term describes the changes in particle density produced by the space-time variation in the rate of addition of particles to this configuration via . The change is brought to by the propagator, . This backflow term in the generalized Boltzmann equation thus describes how information from the collision is brought back to the distribution function, . This term, and its analogs in more accurate approximations,121212For example, Ref  describes how spin is included by making Eq.(4-6a) and Eq.(4-6b) into matrix statements, with anti-commutators on the right and commutators on the left. might be the source of complexity related to the entanglement of quantum states. Usually, we believe that entanglement is connected with phases of complex wave functions. However, because the ’s are real these phases cannot appear directly. Nonetheless a tremendous amount of physical complexity can be encoded in functions of four variables so some functional analog of entanglement might well be possible.
The equations just presented provide a complete theory of the dynamics of degenerate (or non-degenerate) quantum systems slowly varying131313Notice that the “functional analog” of entanglement mentioned in the previous paragraph might well involve the generation of short-range correlations, thereby voiding the conditions required for thre validity of the kinetic equations. in . Relatively simple formulas can be used to find many physical quantities. The simplest example, coming directly from the meaning of the Green’s function, is that the density of particles, momentum, and energy are[page 127] given by or closely related to the integral over of multiplied by, respectively, , and . We shall discuss a few of these connection below. In addition, the Green’s functions yield direct expressions even in non-equilibrium situations for such quantities as correlation energy, stress tensor, and partition function. From these we can derive a non-equilibrium counterpart of temperature times entropy.141414H.C. Öetttinger has also emphasized the necessity for defining temperature in order to pull entropy out of a dynamical argument. However, we have noticed that Boltzmann’s and Landau’s theories permit a simple definition of kinetic entropy even when temperature is undefined. However, as we shall see, entropy151515Recent work [42, 43] describes equilibrium behavior in energy eigenstates without invoking an analysis involving something like an H-theorem itself remains elusive as a general concept in the non-equilibrium part of this Green’s function formulation.
4.2 The Maths
4.2.1 Hilbert transforms
A crucial element of the definitions just given is the Hilbert transform of a function of time, . Given a function with Fourier transform defined by
Let be the function that is plus one for positive and minus one for negative . Now one can see that the Fourier transform of sign is
In Fourier space, then, the Hilbert transform is given by a convolution with . In real space it is defined by multiplication by sign. Thus, if we apply the Hilbert transform twice we get minus our starting point. As a result,