Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 356 (Stochastic Processes in Theoretical Physics and Biology)

Let $w_{ij}$ be the transition rate from $i$ to $j$ and define the probability current $J_{ij}=w_{ij}{P}_i-w_{ji}{P}_j=-J_{ji}$. The master equation is ${\dot{P}}_i=-{\sum }_j{J}_{ij}$. Differentiating the Shannon entropy $S_{\mathrm{sys}}=-{\sum }_i{P}_i\log P_i$ and symmetrizing givesAdding the entropy flow to the environment produces the nonnegative entropy production rate of a Markov chainThe paper's displayed $\sum P\log P$ is the negative of the thermodynamic system entropy, so its derivative has the opposite sign before the environmental contribution is added.

For the two-state chain,with $P_1+P_2=1$.

Eliminating $P_2=1-P_1$ gives ${\dot{P}}_1=\beta -(\alpha +\beta )P_1$. ThereforeWith $r=\alpha p-\beta (1-p)$,

The only independent current isHence the total transient entropy production isIts relaxational part may be writtenwhile the steady or housekeeping entropy production is zero. Both the current and total production vanish as $t\to \infty$.

At stationarity,Thus every edge current vanishes and the two-state system satisfies detailed balance. A network containing only one undirected edge cannot sustain a stationary cycle current.

With indices understood cyclically,The circulant generator has eigenvalues $0$ and $-3\varphi \pm i\sqrt{3}\psi$, where $\varphi =(\alpha +\beta )/2$ and $\psi =(\alpha -\beta )/2$. Expanding the initial vector $(1,0,0)$ in its three Fourier eigenvectors gives

For the oriented edge $i\to j=i+1(\mathrm{mod}3)$, $J_i=\alpha P_i-\beta P_j$. Substitution in the Markov-chain entropy production givesSince ${\sum }_i{J}_i=\alpha -\beta$,The second term is relaxational and vanishes for the uniform stationary distribution. The steady entropy exported to the environment is therefore

At stationarity $P_i=1/3$, but the cycle current is $J_i=(\alpha -\beta )/3$. Therefore the chain is out of detailed balance exactly whenThe nonzero cycle affinity $3\log (\alpha /\beta )$ sustains the stationary current and positive housekeeping entropy production.

Equation (1) is Underdamped Langevin dynamics for a unit-mass particle in potential $U$, coupled to a heat bath of temperature $T$. The coefficient $\gamma >0$ is viscous friction, and the noise amplitude is fixed by the Fluctuation-dissipation theorem. Its Fokker-Planck equation isFor $\gamma \gg 1$, velocity relaxes rapidly, so formallyThus the Overdamped Langevin dynamics isand its density obeys

For $0\le F_0<1$, stationary points satisfy $\cos x=F_0$. There is one minimum $m_k=2\pi -\arccos F_0+2\pi k$ and one maximum $M_k=\arccos F_0+2\pi k$ per period. The potential is a sinusoidal washboard tilted downward to the right by $2\pi F_0$ per period.

Periodization partitions the real line into translated cells, soTranslation by $2\pi$ merely reindexes the sum, proving periodic boundary conditions; summing the Fokker--Planck equation proves that $\hat{p}$ obeys it.

At stationarity the current $J_0=-D{\hat{p}}_s^{\prime }-{\varphi }^{\prime }{\hat{p}}_s$ is constant. Solving this first-order equation and imposing periodicity givesHere $J_0$ is the stationary probability crossing any point per unit time. Normalization determines it:

Each net crossing of a periodic cell advances the unwrapped particle by $2\pi$. The stationary mean velocity is thereforeFor $F_0>0$ the numerator is positive, so motion is on average toward increasing $x$, down the tilted potential.

In the low-temperature regime, the particle rapidly equilibrates near a minimum and only rarely crosses a neighboring maximum. Applying Laplace's method to the exact current formula gives the Kramers escape ratesEach right or left escape changes position by $2\pi$, henceThe reduced continuous-time random walk on minima has off-diagonal transition ratesand diagonal generator entry $W(l|l)=-(k_{+}+k_{-})$. Its residence time in each well is exponentially distributed with rate $k_{+}+k_{-}$, and the next jump is right with probability $k_{+}/(k_{+}+k_{-})$.

An exact Gillespie algorithm for one particle is:

For $N\ll K$, simulate particle identities independently and maintain a priority queue of their next event times. For $N\gg K$, store occupation numbers $n_k$ and use aggregate event rates $n_k{k}_{+}$ and $n_k{k}_{-}$ for each well; one population-level Gillespie event then decrements one $n_k$ and increments its neighbor. This replaces work proportional to particle number by work proportional to the number of occupied wells.

The pair $(X_t,V_t)$ is a hybrid-state process: position is continuous and velocity is discrete. It is Markov because its future law is fixed by the present position and velocity. Position $X_t$ alone is generally non-Markovian, since its future displacement depends on the hidden persistent velocity.

The two transport equations areFor $p=p_{+}+p_{-}$ and $j=s(p_{+}-p_{-})$ they becomeEliminating $j$ gives the closed chemotactic telegraph equation

Take ${\lambda }_0\to \infty$ withfixed, so $s$ and $b$ scale as $\sqrt{{\lambda }_0}$. The rapidly relaxing flux satisfies $j\simeq -D{p}_x+\chi c^{\prime }p$, and thereforeZero stationary flux gives $p_s(x)\propto e^{\chi c(x)/D}$. A stationary probability density on $\mathbb{R}$ exists exactly when this exponential is integrable, in addition to the positivity condition $|c^{\prime }|<{\lambda }_0/b$. The limiting process is

The even pair potential $u$ represents repulsion: because it decreases for positive separation, the force $-u^{\prime }(x_i-x_j)$ pushes particles apart. Marginalizing the $N$-particle Fokker-Planck equation givesUnder the mean-field approximation, propagation of chaos gives $P_2(x_1,x_2)\simeq p(x_1)p(x_2)$. Taking $N\to \infty$ yields the closed nonlocal equationThe convolution term is the collective repulsive drift generated by the population density.