Detailed balance

Detailed balance is a microscopic reversibility condition for stochastic dynamics, central in equilibrium statistical mechanics and in Markov models of relaxation to equilibrium.

Prerequisites: probability measure , expectation , relative entropy (KL) , thermodynamic equilibrium , Markov chain (discrete time) , Markov semigroup (continuous time) .

Definition (discrete time)

Let $(X_n)_{n\ge 0}$ be a Markov chain on a countable state space with transition matrix $P=(P_{ij})$. A probability vector $\pi$ is stationary if $\pi=\pi P$. We say detailed balance holds (or the chain is reversible with respect to $\pi$) if for all states $i,j$,

Equivalently, the stationary edge flow $\pi_iP_{ij}$ is symmetric in $(i,j)$, so there are no steady probability currents.

Definition (continuous time jump process)

For a continuous-time Markov chain with jump rates $q_{ij}\ge 0$ ($i\ne j$) and generator $Q$ (rows sum to $0$), detailed balance with stationary $\pi$ means

This is the natural continuous-time analogue.

TFAE (reversibility / detailed balance)

Assume $\pi$ is stationary.

The following are equivalent:

1. (Detailed balance) $\pi_i P_{ij}=\pi_j P_{ji}$ for all $i,j$ (discrete time), or $\pi_i q_{ij}=\pi_j q_{ji}$ for all $i\ne j$ (continuous time).

2. (Time-reversal of paths) For any finite path $i_0,i_1,\dots,i_n$,

   $$\pi_{i_0} P_{i_0 i_1}\cdots P_{i_{n-1} i_n} ={} \pi_{i_n} P_{i_n i_{n-1}}\cdots P_{i_1 i_0},$$

   i.e. the probability of a trajectory in stationarity equals that of its reversed trajectory.

3. (Self-adjointness in $L^2(\pi)$) The Markov operator $(Pf)(i)=\sum_j P_{ij} f(j)$ satisfies

   $$\langle f, Pg\rangle_\pi = \langle Pf, g\rangle_\pi, \quad \text{where } \langle f,g\rangle_\pi=\sum_i \pi_i f(i)g(i).$$

   (Similarly, the generator is self-adjoint for continuous time.)

4. (No stationary currents) The antisymmetric current $J_{ij}=\pi_iP_{ij}-\pi_jP_{ji}$ (or $J_{ij}=\pi_iq_{ij}-\pi_jq_{ji}$) vanishes for all pairs $(i,j)$.

Global balance vs detailed balance

Stationarity $\pi=\pi P$ (or $\pi Q=0$) is sometimes called global balance; it only forces the net inflow at each state to match the outflow. Detailed balance is stronger: it matches inflow/outflow pairwise for each edge.

Entropy production (interpretation)

Out of equilibrium, stationary currents typically lead to positive entropy production. Detailed balance is the stochastic analogue of equilibrium, connecting naturally to thermodynamic entropy and to monotone decay of relative entropy along dynamics defined by the master equation .