## B) Generalized entropies

### 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:

$$\dot{S} \geq \frac{\dot{Q}}{T}$$

### 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: $$p^\star_m = (g')^{-1} \left(\frac{\alpha+\beta \epsilon_m}{C_f} \right)$$,     $$C_f = f'(\sum_m g(p_m))$$

# requirements



### 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

$$\left[J(w_{mn},p_n)-J(w_{nm},p_m) \right]$$

### 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

## REQUIREMENTS 1-3) IMPLY THAT

$$J(w_{mn},p_n) = \psi( j(w_{mn}) - g'(p_n))$$

### WHERE

$$j$$ - arbitrary function

$$\psi$$ - increasing function

# EXAMPLES

### LINEAR MARKOVIAN DYNAMICS

$$\dot{p}_m = \sum_n (w_{mn} p_n - w_{nm} p_m)$$

$$J_{mn} = w_{mn} p_n = \exp(\log w_{mn} + \log p_n)$$

​$$\Rightarrow g'(p_n) = - \log(p_n)$$

$$\Rightarrow S = - \sum_n p_n \log p_n$$

### 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:

$$S = \frac{1}{1-q} (\sum_m p_m^q-p_m)$$

$$\Rightarrow g'(p_m) = \frac{q p_m^{q-1}-1}{1-q}$$

$$J_{mn} = \psi(j(w_{mn}) + \frac{q p_m^{q-1}-1}{q-1} )$$

# SKETCH OF PROOF

$$\dot{S} = - \sum_m \dot{p}_m \log p_m$$

$$= - \frac{1}{2}\sum_{mn} (w_{mn}p_n - w_{nm}p_m) \log \frac{p_m}{p_n}$$

$$= \underbrace{\frac{1}{2}\sum_{mn} (w_{mn}p_n - w_{nm}p_m) \log \frac{w_{mn} p_n}{w_{nm} p_m}}_{\dot{S}_i}$$

$$+ \underbrace{\frac{1}{2}\sum_{mn} (w_{mn}p_n - w_{nm}p_m) \log \frac{w_{mn} }{w_{nm} }}_{\dot{S}_e}$$

$$= \dot{S}_i + \dot{S}_e \geq \frac{\dot{Q}}{T}$$

### 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:

$$j(w_{mn}) - j(w_{nm}) = \frac{\epsilon_n - \epsilon_m}{C_f T}$$

$$\beta = \frac{1}{T}$$

By Jan Korbel

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