Entropy is in Flux V3.4

Abstract

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 systems 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 (Phys Rev 115:1342–1373, 1959) as generalized Wigner functions, $g^<(\mathbf {p},\omega ,\mathbf {R},T)$ and $g^>(\mathbf {p},\omega ,\mathbf {R},T)$. This paper describes once again 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 definition of the density of kinetic entropy. However, when the system is outside of local equilibrium, this definition fails. It is speculated that quantum entanglement is the source of this failure.

Appendices

Appendix 1: $\Phi$-Derivability

Baym [59] showed that one could obtain models that included conservation laws as well as properties of thermodynamic consistency by using approximations that he described as $\Phi$-derivable. (See also the earlier [65].) These approximations used a thermodynamic Green’s function, G(1, 2), where the numerals stand for space–time coordinates.

One starts by constructing $\Phi$ as defined by integrals of G(1, 2) and the two-body potential, V(1, 2). This potential is proportional to delta functions of its time difference variable. A typical structure that appears under the integral signs is

$$\begin{aligned} G(1,a)G(a,1')V(a,b) G(2,b) G(b,2') \text {~with~} V(a,b)=\delta (t_a-t_b)v(\mathbf {R}_1,\mathbf {R}_b) \end{aligned}$$

In an infinitesimal range of the time variables one would get a structure like

$$\begin{aligned} \text {Pr} \int ~d\omega _1~d\omega _{1'} d\omega _2~d\omega _{2'}~ \frac{G^>(\omega _1)G^<(\omega _{1'})v() G^>(\omega _{2'}) G^<(\omega _{2'})}{\omega _1-\omega _{1'}+\omega _2-\omega _{2'}} \end{aligned}$$

(74)

Here I have left out space variables and chosen particular values of the signs of it difference variables. Note the lack of i epsilons in the frequency denominator. These $i\epsilon$ specifications are only necessary when some time integration is taken to infinity and here all integrations are over finite domains. Even the principle value sign is irrelevant because whenever there is a zero in the denominator the “boundary condition”

(75)

ensures that there is a zero in the numerator to cancel out the zero in the denominator. The result further simplifies if V(1, 2) and G(1, 2) are functions of spatial difference variables, as one might get with spatial periodic boundary conditions.

Once one has put together all these integrals, one has constructed a functional, $\Phi [G^>(.),G^<(.),V(.)]$ where . stands either for the space–time difference variable or for the Fourier transform variables $\mathbf {p},\omega$. In addition, and most important, the functional contains a large collection of frequency denominators like the ones in Eq. (74). The theory says that the variational derivative of $\Phi$ obeys

$$\begin{aligned} \frac{\delta \Phi }{\delta G(1,2)} = \Sigma (1,2) \end{aligned}$$

(76)

Since both G and $\Sigma$ have a discontinuous jump when their time variables become equal to one another, one could say in more detail that

$$\begin{aligned} \frac{\delta \Phi }{\delta G^>(1,2)} =-i \Sigma ^<(1,2) \text {~~for~~} it_1<it_2 \end{aligned}$$

(77a)

while

$$\begin{aligned} \frac{\delta \Phi }{\delta G^<(1,2)} =-i \Sigma ^>(1,2) \text {~~for~~} it_2<it_1 \end{aligned}$$

(77b)

The factors of $-i$ are inserted for later convenience in using $\Phi$. To be consistent with Eq. (77), derivatives in the $\mathbf {p},\omega$ representation must have the value

$$\begin{aligned} \frac{\delta \Phi }{\delta G^>(\mathbf {p},\omega )} =\text {Pr} \int \frac{d\omega '}{\pi }~ \frac{\Sigma ^<(\mathbf {p},\omega ')}{\omega -\omega '} \end{aligned}$$

(78a)

while

$$\begin{aligned} \frac{\delta \Phi }{\delta G^<(\mathbf {p},\omega )} =-\text {Pr} \int \frac{d\omega '}{\pi } \frac{\Sigma ^>(\mathbf {p}.\omega ')}{\omega -\omega '} \end{aligned}$$

(78b)

The frequency denominators arise from the ubiquitous denominators sitting in $\Phi$, as described above. Since

$$\begin{aligned} \text {Pr} \int \frac{d\omega '}{\pi }~\frac{\exp {(-i\omega 't)}}{\omega -\omega '} = -\text {sign}(t) i \end{aligned}$$

Eq. (78) agrees with Eq. (77).

Note that $\Phi$ is an extensive quantity. One of the many space–time integrals in it gives the volume of the system $\Omega$ multiplied by the time interval $-i\beta$ so that

$$\begin{aligned} \Phi [G^>,G^<]= -i~\beta ~ \Omega ~ \phi [g^>,g^<] \end{aligned}$$

The g’s on the right are functions of $\mathbf {p}$ and $\omega$ such that

$$\begin{aligned} \frac{\delta \phi }{\delta g^>(\mathbf {p},\omega )} =\text {Pr} \int \frac{d\omega '}{\pi }~ \frac{\sigma ^<(\mathbf {p},\omega ')}{\omega -\omega '} \end{aligned}$$

(79a)

while

$$\begin{aligned} \frac{\delta \phi }{\delta g^<(\mathbf {p},\omega )} =-\text {Pr} \int \frac{d\omega '}{\pi } \frac{\sigma ^>(\mathbf {p}.\omega ')}{\omega -\omega '} \end{aligned}$$

(79b)

Appendix 2: Entropy Correction

The correction to the perfect integral appearing in the final formula of the text, Eq. (73c) is

$$\begin{aligned} \Delta =\int ~dp~ \Big [ \frac{-\sigma ^<}{f} + \frac{\sigma ^>}{1+\varsigma f} \Big ] [g,f] =\int ~dp~ \frac{[-\sigma ^< g^>+\sigma ^> g^<]}{ a f (1+\varsigma f)} [g,f] \end{aligned}$$

(80)

This expression is second order in a gradient expansion since every square bracket encloses a expression that is zero in global equilibrium.

The final expression in Eq. (80) is just of the right form to use a variation of the variational replacement idea of Sect. 4.4 with

$$\begin{aligned} -O^>(p)=O^<(p)=\frac{ [g(p),f(p)] }{ a(p) ~f(p)~[1+\varsigma f(p)]} \end{aligned}$$

(81)

Then the correction term comes out in a form based upon $\psi [g^<,g^>]$ that is

$$\begin{aligned} \Delta= & {} \frac{1}{2} \int ~dp ~dq~ dp' ~dq' ~ [O^<(p)+O^<(q)-O^<(p')-O^<(q')] \nonumber \\&\mathcal {Q} \left( \begin{array} {ccc} p &{} \rightarrow &{} p' \\ q &{} \rightarrow &{} q' \\ \end{array}\right) \delta (\mathbf {p}+\mathbf {q}-\mathbf {p}'-\mathbf {q}') \nonumber \\&\{ g^<(p) ~g^<(q)~ g^>(p') ~g^>(q')\} \end{aligned}$$

(82)

Here, as in Boltzmann’s analysis of his collision term we see that most of the expression has a simple parity under the interchange of primed and unprimed variables. Up to the last line, everything is odd under the interchange. To make the whole expression even, we antisymmetrize the last line and find that

$$\begin{aligned} \Delta= & {} \frac{1}{4} \int ~dp ~dq~ dp' ~dq' ~ [O^<(p)+O^<(q)-O^<(p')-O^<(q')] \nonumber \\&\mathcal {Q} \left( \begin{array} {ccc} p &{} \rightarrow &{} p' \\ q &{} \rightarrow &{} q' \\ \end{array}\right) \delta (\mathbf {p}+\mathbf {q}-\mathbf {p}'-\mathbf {q}') \nonumber \\&\{ g^<(p) ~g^<(q)~ g^>(p') ~g^>(q')- g^<(p') ~g^<(q')~ g^>(p) ~g^>(q)\} \end{aligned}$$

(83)

I have not been able to further simplify this expression.