Nonequilibrium entropy: Can we define it unambiguously

1
Nonequilibrium entropy Can we define it
unambiguously?
Takafumi KITADepartment of Physics, Hokkaido
University , Sapporo
Entropy

• The key quantity in thermodynamics and
  statistical mechanics

Clausius inequality the law of increase of
entropy

• No definition in nonequilibrium (except the one
  by Boltzmann)

Cannot trace nonequilibrium time evolution of
entropy
Derive nonequilibrium expression of entropy with
QFT
2
Boltzmanns two definitions of entropy
(1) 1872 (Boltzmann equation)
dilute classical gas, applicable to nonequilibrium
(2) 1877 (Boltzmanns principle)
equilibrium, no time evolution (cf von Neumann)
3
Deriving nonequilibrium entropy

• use quantum field theory
• follow Boltzmanns derivation of entropy in
  1872
• ask compatibility with equilibrium statistical
  mechanics

4
Development of transport theoryand problems to
be resolved
5
Boltzmann eq.(1872)
distribution in phase space
time evolution
(C collision integral)
net change of f caused by collisions
and
Substitution of
evolution equation of nonequilibrium systems
6
H theorem(1872)
by
multiplying
entropy density
entropy flux density
dS/dt ?0 when
7
Can only handle dilute classical gases
Two-particle collisions, etc.

• Incorporating strong correlations systematically

Evolution of f sufficinent?

• Quantum effects

8
Study by Kadanoff and Baym (1962)
field-theoretic
approach (Matsubara Greens function
analytic continuation)
Partition function
Complex t plane
Matsubara Greens fn.(1955)
(considering fermions below)
9
Basic techniques

• Dysons equation

• Wigner transform

• Moyal product

Wigner transform of
10
ex
Different evolutions !?

• left-right

• e- integration

11
Questions
t plane
(a) Imaginary time analytic continuation
Cant we trace real-time evolutions ?
(b) Approximation
left-right subtraction trick
Lost some information?
Needs quasiparticle approx. in the presence of
interaction
does not vanish with partial integration
12
Real-time perturbation by Keldysh(1964)
L. V. Keldysh Sov. Phys. JETP 20 (1965) 1018
Expectation value of operator
13
Perturbation expansion on contour C
(the same technique as the equilibrium one)
Feynman diagrams and Dysons equation on C
Need distinguish the forward and backward paths
14
Keldysh Greens function
?
independent
15
Keldysh rotation and derivation of transport
equation
The Kadanoff-Baym procedure for
(gradient expansion left-right quasiparticle
approx.)
?
Boltzmann equation for electrons
16
To be performed
T. Kita J. Phys. Soc. Jpn. 75 (2006) 114005
(1) Deriving quantum transport equations
systematically
Systematic approximation scheme
(2) Deriving an expression of nonequilibrium
entropy
Time evolution of entropy(impossible with
thermodynamics)
cf Ivanov et al., Nucl. Phys. A 672 (2000) 313.
17
Deriving quantum transport equation and
nonequilibrium entropy
18
System
Interacting fermions (bosons)
19
Interaction representation
An extra contribution from C2 (potential -V)
20
Dysons equation
self-energy
21
F-derivative approximation
Satisfying various conservation laws automatically
22
Wigner transform
Fourier transform with respect to relative
coordinates
Expressing independent Fourier coefficients as
Spectral function A
23
Wigner transform of self-energy
Fourier transform with respect to relative
coordinates
Expressing independent Fourier coefficients as
24
Keldysh rotation
Independent elements
GR and GA expressible in terms of A only
Both A and f necessary for GK
(cf only A is unknown in equilibrium)
25
Moyal product and gradient expansion
generalized Poisson bracket
Microscopic length, time ltlt characteristic
scales of space-time inhomogeneity
26
Approximation
Botermans and Malfliet Phys. Rep. 198 (1990) 115.
Require that the two equations below be identical
(1) left Dyson - right Dyson0 (2) left Dyson
right Dyson2
To be specific (1) first-order gradient
expansion (2) fS?f in space-time derivatives
27
Equations for A and f
(a) Equation for A (determining density of states)
(b) Equation for f (determining distribution)
collision integral
A set of two coupled equations for A and f
(cf A only in equilibrium f only in Boltzmann
eq.)
28
Including Boltzmann eq. as dilute classical limit
Equation for f ?
29
Expression of nonequilibrium entropy (1)
Multiply equation of f by
integrate over pe
30
Expression of nonequilibrium entropy(2)
An expression embracing equilibrium entropy T.
Kita J. Phys. Soc. Jpn. 68 (1999) 3740.
Equilibrium
31
Entropy by Ivanov et al.
Ivanov et al. Nucl. Phys. A 672 (2000) 313.
smemory contribution of gradient terms in C
?
Reproducing Carneiro-Pethick entropy in
equilibrium
Carneiro and Pethick Phys. Rev. B 11 (1975)
1106.
Carneiro-Pethick entropy not correct,
however! (inappropriate treatment of energy
denominators)
32
H theorem
Integration of
?

• Can be proved within second order
• expected to hold generally

when
(isolated)
33
Summary

• Deriving quantum transport equations with
  the many-body quantum field theory (rigorous
  derivation of Boltzmann equation and its
  extension)
• Expression of nonequilibrium entropy and its
  evolution equation

Enabling microscopic (statistical-mechanical) desc
ription of nonequilibrium time evolutions
34
Related topics

• thermal perturbation

Description with the Enskog expansion

• two-body correlations

Description with the F-derivative approximation

• charged system

Adequate treatment of gauge invariance ?
description of Hall effect
T. Kita Phys. Rev. B 64 (2001) 054503

• Extremal principle in nonequilibrium systems !?

35
Thermal disturbance
Enskog expansion from local equilibrium
and
to be determined with conservation laws
36
Two-body correlations
Bethe-Salpeter equation for
where G is defined by
K can be calculated once F is given
37
Gauge invariance and Hall terms
The folloing Wigner transformation is not
appropriate for charged systems!

• gauge invariance violated

(2) absence of Hall terms
38
Modified Wigner transformation
Stratonovich(1956), Fujita (1966)
Derivation of the corresponding transport eq.
Levanda and Fleurov J. Phys. C. M. 6 (1994)
7889.
T. Kita Phys. Rev. B 64 (2001) 054503.
39
Moyal product
T. Kita Phys. Rev. B 64 (2001) 054503.
40
comment alternative method
Keldysh(1964), Kubo(1964), Altshuler(1978),

• violation of gauge invariance except static cases

Stratonovich(1956), Serimaa et al.(1986)

• not applicable to superconductors

41
Extremal principle in nonequilibrium?

• Can describe nonequilibrium time evolutionsin
  terms of A and f

cf Tsallis statistics

• Is there a functional of A and f which becomes
  extremal in steady states?

How about entropy?

• continuous description from equilibrium
• valid up to first order in deviation from
  local equilibrium?

42
Entropy change through Rayleigh-Bénard
convective transition
convective state
conductive state
heat flux
Temperature variation
43
Rayleigh number R?gad3DT/kn
g standard gravity, a thermal expansion
coeff., k heat diffusion coeff., n dynamic
viscosityd width of the system, DT
temperature difference
Observation of Nusselt number(Silveston1958)
Nonequilibrium phase transition!
44
Entropy change through the transition

• candidates roll, square, hexagonal

• initial conditionconduction fluctuations(RgtRc)

• trace entropy change in time

Entropy increasing monotonically towards a
constant
45
Summary

• Development of nonequilibrium transport theory
• probability distribution in nonequilibrium
  systems
• real-time evolution of the distribution
• entropy

No fundamental difficulties in describing
nonequilibrium systems!?