Thermodynamics, fluctuations, and
response for systems out of
equilibrium
Shin-ichi Sasa
(University of Tokyo)
2007/11/05
in collaboration with
T.S. Komatsu, N. Nakagawa, and H. Tasaki
Outline of my talk
1.
2.
3.
4.
Introduction (7min)
Question (4min)
Result (14min)
Conclusion (1min)
2
(Near and in) Equilibrium
Second law
Thermodynamic relation
Fundamental limitation of operations
Unified description of material properties
dU=TdS-pdV
Entropy
Thermodynamic function
Large deviation
Macroscopic fluctuations
Fluctuation-dissipation relation
Linear response formula
Equilibrium distribution
Detailed-balance
expression in terms of “energetic quantities”
Microscopic reversibility
The principle of equal weight
Microscopic equation
3
Non-equilibrium steady state
？
？
Second law
Thermodynamic relation
Fundamental limitation of operations
Unified description of material properties
？
dU=TdS-pdV
Entropy
Thermodynamic function
Large deviation
Macroscopic fluctuations
？
Violation of Fluctuation-dissipation
relation
X
Stationary distribution
Detailed-balance
expression in terms of “energetic quantities”
Microscopic reversibility
The principle of equal weight
Microscopic equation
4
Our results
(KNST=Komatsu, Nakagawa, Sasa and Tasaki)
Second law
Thermodynamic relation
Fundamental limitation of operations
Unified description of material properties
(Sasa-Tasaki, JSP, 2006)
(Hatano-Sasa, PRL, 2001)
Entropy
(Sasa,arXiv0706.0043)
Large deviation
(KNST, arXiv0711.0426)
(Harada-Sasa, PRL, 2005)
A formula for the violation of FDR
Macroscopic fluctuations
Valid up to O(e
Stationary distribution
Local detailed-balance
expression in terms of “energetic quantities”
Microscopic reversibility
(Komatsu-Nakagawa, arXiv0708.3158)
e : the degree of non-equilibrium
(KNST, 2007)
Microscopic equation
5
Outline of my talk
1.
2.
3.
4.
Introduction (7min)
Question (4min)
Result (14min)
Conclusion (1min)
6
Heat conducting steady state
piston
Heat bath
Stochastic system
(Hamiltonian system)
and other …..
(volume)
system
Heat bath
Hamiltonian system
Stochastic system
(Hamiltonian system)
and other …..
Microscopic description
Set of parameters in the Hamiltonian
Control parameter
7
Equilibrium case
8
Question
• Existence of “Thermodynamic function”
F(T,n
in non-equilibrium steady state ?
Y characterizes the non-equilibrium nature
T “temperature”
(Y  const .)
- These should be determined operationally.
- It should contain new predictions that can be checked
experimentally
9
difficulty
• To seek for such a framework is a danger
project, because there is much ambiguity.
“What is T? “ “ What is Y? “
• Careful arguments (with proposing a possible
form) were presented in the paper, Sasa and
Tasaki, JSP, 2006 (100 pages!)
Outline of my talk
1.
2.
3.
4.
Introduction (7min)
Question (4min)
Result (14min)
Conclusion (1min)
11
Heat conducting steady state
piston
Heat bath
Stochastic system
(Hamiltonian system)
and other …..
(volume)
system
Heat bath
Hamiltonian system
Stochastic system
(Hamiltonian system)
and other …..
Microscopic description
Set of parameters in the Hamiltonian
Control parameter
12
Useful representation
Komatsu-Nakagawa
arXiv0708.3158
dimensionless
heat flux
Conditional path ensemble average
G
Energy into the left heat bath
Energy into the right heat bath
13
Remark
Equilibrium case
S : thermodynamic entropy
14
Significance
- Universal form for many non-equilibrium systems
- Linear response formula (in an elegant manner)
- Non-linear response formula
- Large deviation functional
- Steady state thermodynamics
15
Clausius’ formula
Quasi-static protocol
Equilibrium case
Quasi-static heat
out of the baths
Non-equilibrium case
16
Extended Clausius’ formula
Quasi-static reverse protocol
(KNST, preprint, arXiv:0711.0426)
17
Thermodynamics
Heat bath
Stochastic system
(Hamiltonian system)
and other …..
(volume)
system
Hamiltonian system
(KNST, in preparation)
Heat bath
Stochastic system
(Hamiltonian system)
and other …..
18
Result
( J  const .)
e.g.
( J  const .)
pressure in the J direction
The bulk region is changed only in the J direction
Existence of Steady State Thermodynamics !
(KNST, preprint, arXiv:0711.0246)
19
Significance I
• Universal macroscopic theory based on microscopic
mechanics !
• Non-trivial relation among experimentally
measurable quantities (through the Maxwell
relation)
e.g.
Compressibility tensor in a heat conducting fluid
• All theories that discuss statistical behavior beyond
the linear response regime must satisfy our relation
e.g. Check the validity of the Boltzmann equation
Significance II
• New physics in terms of
• Mechanical detection of the departure from
the local equilibrium (such as “long range
correlation”)
• Enlightening guide in a (future) construction of nonequilibrium statistical mechanics !
21
Summary
(KNST=Komatsu, Nakagawa, Sasa and Tasaki)
Second law
Thermodynamic relation
Fundamental limitation of operations
Unified description of material properties
(Sasa-Tasaki, JSP, 2006)
(Hatano-Sasa, PRL, 2001)
Entropy
(Sasa,arXiv0706.0043)
Large deviation
(KNST, arXiv0711.0426)
(Harada-Sasa, PRL, 2005)
A formula for the violation of FDR
Macroscopic fluctuations
Valid up to O(
Stationary distribution
Local detailed-balance
expression in terms of “energetic quantities”
Microscopic reversibility
(Komatsu-Nakagawa, arXiv0708.3158)
 : the degree of non-equilibrium
(KNST, 2007)
Microscopic equation
22