Questions: Master Equation

Question 1

A stochastic system reaches a state where dP_n/dt = 0 for all states n. A student concludes the system is in thermodynamic equilibrium because it satisfies detailed balance. What is wrong with this inference?

• A. Steady state is impossible in Markovian systems — the system is always evolving
• B. Detailed balance is a sufficient but not necessary condition for steady state: a system can have dP_n/dt = 0 in a non-equilibrium steady state where probability flows in cycles without W_{nm}P_m = W_{mn}P_n for every pair
• C. The student is correct — steady state always implies detailed balance in a Markovian system
• D. Detailed balance requires that the system be in contact with a thermal bath, which may not hold here

Answer: (B)

This is the key distinction between equilibrium and non-equilibrium steady states. Steady state (dP_n/dt = 0) requires only that the total probability flowing into each state equals the total flowing out — probability can still circulate in loops without pairwise balance. Detailed balance is stronger: it requires that the flow from state n to m exactly equals the flow from m to n for EVERY pair. Detailed balance implies equilibrium; steady state does not. Biological motors and driven systems can maintain steady states far from equilibrium without satisfying detailed balance.

Question 2

In the master equation dP_n/dt = Σ_m [W_{nm}P_m − W_{mn}P_n], the gain term W_{nm}P_m represents:

• A. The rate at which the system transitions from state n to all other states m
• B. The equilibrium probability of state n, weighted by the transition rate
• C. The rate of probability flowing into state n from state m, equal to the current probability of being in m multiplied by the transition rate from m to n
• D. The probability of state n at steady state, determined by the ratio of forward and reverse rates

Answer: (C)

The gain term W_{nm}P_m accounts for all transitions INTO state n from any other state m. W_{nm} is the rate of transitioning from m to n; P_m is the probability of currently being in m. Their product gives the rate at which probability flows from m into n. Summing over all m gives the total gain rate for state n. The loss term W_{mn}P_n symmetrically accounts for all transitions OUT of n. This gain-minus-loss structure is the balance equation — it is physically transparent and holds for any Markovian system.

Question 3

The Markov property means that transition rates W_{nm} depend on the full history of states the system has visited, not just the current state.

• T. True
• F. False

Answer: False

The Markov property means the OPPOSITE: transition rates depend only on the current state, not on the history. This is the memoryless assumption — the system has 'forgotten' how it arrived at its current state before the next transition occurs. Physically, this holds when the environment relaxes much faster than the transition timescale, so no information about the past is encoded in the current configuration. Non-Markovian dynamics (where history matters) require more complex formalisms beyond the master equation.

Question 4

In the limit where discrete states become densely packed (continuum limit), the master equation reduces to the Fokker-Planck equation for continuous probability distributions.

• T. True
• F. False

Answer: True

This connection confirms that the master equation and Fokker-Planck equation describe the same physical content in different regimes. When the state space is discrete and widely spaced, the master equation is the natural formalism. When states become densely packed and transitions are small, expanding the transition rates in a Taylor series and taking the continuum limit yields the Fokker-Planck equation — a partial differential equation for a continuous probability density. The two formalisms are complementary tools for the same class of Markovian stochastic processes.

Question 5

Explain what the Markov property physically means, and give an example of a physical system where it is a reasonable approximation and explain why.

Model answer: The Markov property means that the future evolution of a system depends only on its current state, not on its past history — the system is memoryless. This is reasonable when the timescale of environmental fluctuations (the 'bath') is much shorter than the timescale of state transitions, so by the time a transition occurs, the environment has fully relaxed and no memory of the past is retained. Example: a molecule deciding whether to isomerize (change conformation) in solution. The surrounding solvent molecules collide and thermalize on picosecond timescales, much faster than the microsecond-to-millisecond timescale of conformational changes. By the time isomerization occurs, the molecule has no 'memory' of its collision history.

The Markov assumption is an approximation that fails when environmental relaxation is slow relative to the system's transitions — this generates non-Markovian (memory) effects. But for many physical, chemical, and biological systems, the bath relaxes quickly and the approximation is excellent. It is also philosophically important: it means the entire future probability distribution is determined by the current distribution alone, making the master equation a closed equation for P_n(t).