Second law, detailed balance and linear Markovian dynamics determine Shannon entropy
Jan Korbel and David H. Wolpert
"stochastic thermodynamics in complex systems"
csh online workshop
28th May, 2020
In this talk, we will explore THE relationship between two aspects of thermodynamics:
A) Stochastic thermodynamics
B) Generalized entropies
A) Stochastic thermodynamics
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emergent field of thermodynamics (since 90's)
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Describes non-equlibrium thermodynamics by stochastic variables, especially in microscopic systems
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Main results (other talks): fluctuation theorems, thermodynamic uncertainty relations, nanomotors...
A) Stochastic thermodynamics
key aspects
1) Master equation: Linear Markovian dynamics
\( \dot{p}_m = \sum_n (w_{mn} p_n - w_{nm} p_m) \)
2) (LOCAL) detailed balance: Probability Currents Vanish FOR (LOCAL) EQUILIBRIUM DISTRIBUTIONS
\( \frac{w_{mn}}{w_{nm}} = \frac{p^\star_m}{p^\star_n} = \exp\left(-\frac{\epsilon_m-\epsilon_n}{T} \right)\)
3) Second law of thermodynamics:
\( \dot{S} \geq \frac{\dot{Q}}{T} \)
B) generalized entropies
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STUDIED IN INFORMATION THEORY SINCE 60'S
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used in physics since 90's
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Main aim: study thermodynamics of systems with non-botlzmannian equilibrium distributions (due to correlations, long-range interactions...)
B) generalized entropies
key aspects
I.) general form of entropy:
\( S(P) = f\left(\sum_m g(p_m) \right) \)
II.) Maximum entropy principle:
Maximize S(p) subject to constraint that p is normalized and expected energy has a given value
Solution: MaxEnt distribution: \( p^\star_m = (g')^{-1} \left(\frac{\alpha+\beta \epsilon_m}{C_f} \right) \), \( C_f = f'(\sum_m g(p_m)) \)
QUestion: For what general form of entropies do the key aspects of stochastic thermodynamics hold if the system is off equilibrium?
requirements
blue - standard Stochastic thermodynamics
0) Definitions
\( \)
internal energy
\( U = \sum_m p_m \epsilon_m\)
entropy
\( S = f\left(\sum_m g(p_m) \right) \)
\( S = -\sum_m p_m \log p_m \)
1) Markovian Dynamics
\( \dot{p}_m = \sum_n \left[J(w_{mn},p_n)-J(w_{nm},p_m) \right] \)
\( \dot{p}_m = \sum_n (w_{mn} p_n - w_{nm} p_m) \)
normalization
\( \sum_m \dot{p}_m = 0\)
transition rates
\( w_{mn}\)
probability CURRENTS
\( \left[J(w_{mn},p_n)-J(w_{nm},p_m) \right]\)
2) DETAILED BALANCE
Two ways how to characterize equilibrium:
a) Maximum entropy principle
\(p^\star_m = (g')^{-1} \left( \frac{\alpha+\beta \epsilon_m}{C_f} \right) \)
\(p^\star_m = \exp(-\alpha-\beta \epsilon_m) \)
B) PROBABILITY CURRENTS VANISH
\( J(w_{mn},p^\star_n)=J(w_{nm},p^\star_m) \)
\(w_{mn} p^\star_n = w_{nm} p^\star_m\)
3) Second law of thermodynamics
\(\frac{\mathrm{d} S}{\mathrm{d} t} = \dot{S}_i + \dot{S}_e\)
Entropy production RATE
\( \dot{S}_i \geq 0\) and \(\dot{S}_i = 0 \Leftrightarrow J(w_{mn},p_n) = J(w_{nm},p_m) \ \forall \ m,n\)
Entropy flow rate
\( \dot{S}_e = \frac{1}{T} \sum_m \dot{p}_m \epsilon_m = \frac{\dot{Q}}{T} \)
MAIN RESULT
THEOREM:
REQUIREMENTS 1-3) IMPLY THAT
\(J(w_{mn},p_n) = \psi( j(w_{mn}) - g'(p_n)) \)
WHERE
\(j\) - arbitrary function
\(\psi\) - increasing function
IDEA OF THE PROOF
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CALCULATE TIME DERIVATIVE OF ENTROPY
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DIVIDE IT INTO
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NON-NEGATIVE ENTROPY PRODUCTION RATE
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ENTROPY FLOW RATE
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USE DETAILED BALANCE
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FROM ENTROPY FLOW RATE WE GET CONSTRAINTS ON THE FORM OF THE CURRENT
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PROOF IN THE APPENDIX (AVAILABLE ON WEB)
EXAMPLES
LINEAR MARKOVIAN DYNAMICS
\( \dot{p}_m = \sum_n (w_{mn} p_n - w_{nm} p_m) \)
\(J_{mn} = w_{mn} p_n = \exp(\log w_{mn} + \log p_n)\)
\(\Rightarrow g'(p_n) = - \log(p_n) \)
\( \Rightarrow S = - \sum_n p_n \log p_n\)
Requiring second law, detailed balance, and linear
Markovian dynamics forces entropy to be Shannon entropy
Finite heat Bath
Hamiltonian:
\(H = H_{system} +H_{bath} \)
SCALINg:
\( \lambda \, H_{bath}(x_1,\dots,x_n) = H_{bath}( \lambda^{1/a_1} x_1,\dots, \lambda^{1/a_n} x_n) \)
EQUILIBRIUM:
\( p(E) \propto \int \delta(E - H_{bath}) \, \mathrm{d} x_1 \dots \mathrm{d} x_n \)
q-exp:
\( p(E) \propto (1-(q-1) \beta E)^{1/(q-1)} \)
Tsallis entropy:
\( S = \frac{1}{1-q} (\sum_m p_m^q-p_m)\)
\( \Rightarrow g'(p_m) = \frac{q p_m^{q-1}-1}{1-q}\)
\( J_{mn} = \psi(j(w_{mn}) + \frac{q p_m^{q-1}-1}{q-1} )\)
Master equation:
Finite heat Bath
CONSEQUENCES
REASONABLE SCENARIOS
IF ALL REQUIREMENTS ARE OBEYED
SYSTEM'S DYNAMICS IS non-linear
IF ALL REQUIREMENTs except 1) ARE OBEYED
SYSTEM'S DYNAMICS IS NON-MARKOVIAN
Finite heat Bath
CONSEQUENCES
UNREASONABLE SCENARIOS
IF ALL REQUIREMENTS EXCEPT 2) ARE OBEYED
THEN THE DISTRIBUTION OBTAINED FROM ENTROPY MAXIMIZATION WOULD BE A NON-EQUILIBRIUM STEADY STATE
IF ALL REQUIREMENTS EXCEPT 3) ARE OBEYED
THEN SECOND LAW OF THERMODYNAMICS WOULD BE VIOLATED
MAIN IDEA
NON-BOLTZMANNIAN EQUILIBRIUM DISTRIBUTION
IN A system satisfying
detailed balance and 2nd law
forces the system to obey either
non-linear or non-Markovian DYNAMICS
APPENDIX
SKETCH OF PROOF
\( \dot{S} = - \sum_m \dot{p}_m \log p_m \)
\(= - \frac{1}{2}\sum_{mn} (w_{mn}p_n - w_{nm}p_m) \log \frac{p_m}{p_n}\)
\(= \underbrace{\frac{1}{2}\sum_{mn} (w_{mn}p_n - w_{nm}p_m) \log \frac{w_{mn} p_n}{w_{nm} p_m}}_{\dot{S}_i}\)
\(+ \underbrace{\frac{1}{2}\sum_{mn} (w_{mn}p_n - w_{nm}p_m) \log \frac{w_{mn} }{w_{nm} }}_{\dot{S}_e}\)
\(= \dot{S}_i + \dot{S}_e \geq \frac{\dot{Q}}{T}\)
Standard stochastic thermodynamics
SKetch of proof
\( \dot{S} = C_f \sum_m \dot{p}_m g'(p_m) \)
\(= \frac{C_f}{2} \sum_{mn} (J_{mn}-J_{nm}) (g'(p_m) - g'(p_n)) \)
\( = \frac{C_f}{2} \sum_{mn} (J_{mn}-J_{nm}) (\Phi_{mn} - \Phi_{nm}) \)
\(+ \frac{C_f}{2} \sum_{mn} (J_{mn}-J_{nm}) (g'(p_m)+\Phi_{nm} - g'(p_n) - \Phi_{mn}) \)
\( = \underbrace{\frac{C_f}{2} \sum_{mn} (J_{mn}-J_{nm}) (\phi(J_{mn}) - \phi(J_{nm}) )}_{\dot{S}_i}\)
\(+ \underbrace{\frac{C_f}{2} \sum_{mn} (J_{mn}-J_{nm}) (g'(p_m)+\phi(J_{nm}) - g'(p_n) - \phi(J_{mn}) )}_{\dot{S}_e} \)
SKetch of proof
\(\dot{S}_i \Rightarrow \phi - \ increasing\)
\(\dot{S}_e \Rightarrow C_f[g'(p_m) + \phi(J_{nm}) - g'(p_n) - \phi(J_{mn})] = \frac{\epsilon_n - \epsilon_m}{T} \)
\(\Rightarrow \phi(J_{mn}) = j(w_{mn}) - g'(p_n) \)
\(\Rightarrow J_{mn} = \psi(j(w_{mn}) - g'(p_n))\), \( \psi = \phi^{-1} \) - increasing \( \square . \)
Notes:
\( j(w_{mn}) - j(w_{nm}) = \frac{\epsilon_n - \epsilon_m}{C_f T} \)
\( \beta = \frac{1}{T} \)
analogous for multiple heat baths
CSH Online Workshop "Stochastic thermodynamics in complex systems"
By Jan Korbel
CSH Online Workshop "Stochastic thermodynamics in complex systems"
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