Lecture

Markov Chains: Reversibility and Irreducibility

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Description

This lecture covers the concepts of reversibility and irreducibility in Markov chains, focusing on the conditions for a chain to be reversible and the implications of irreducibility. The instructor explains the detailed balance conditions, transition probabilities, and the optimal policy for decision-making in a Markov chain scenario.

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Related concepts (32)

Markov chain

A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs now." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC).

Microscopic reversibility

The principle of microscopic reversibility in physics and chemistry is twofold: First, it states that the microscopic detailed dynamics of particles and fields is time-reversible because the microscopic equations of motion are symmetric with respect to inversion in time (T-symmetry); Second, it relates to the statistical description of the kinetics of macroscopic or mesoscopic systems as an ensemble of elementary processes: collisions, elementary transitions or reactions.

Detailed balance

The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). It states that at equilibrium, each elementary process is in equilibrium with its reverse process. The principle of detailed balance was explicitly introduced for collisions by Ludwig Boltzmann. In 1872, he proved his H-theorem using this principle. The arguments in favor of this property are founded upon microscopic reversibility.

Time reversibility

A mathematical or physical process is time-reversible if the dynamics of the process remain well-defined when the sequence of time-states is reversed. A deterministic process is time-reversible if the time-reversed process satisfies the same dynamic equations as the original process; in other words, the equations are invariant or symmetrical under a change in the sign of time. A stochastic process is reversible if the statistical properties of the process are the same as the statistical properties for time-reversed data from the same process.

Reversible process (thermodynamics)

In thermodynamics, a reversible process is a process, involving a system and its surroundings, whose direction can be reversed by infinitesimal changes in some properties of the surroundings, such as pressure or temperature. Throughout an entire reversible process, the system is in thermodynamic equilibrium, both physical and chemical, and nearly in pressure and temperature equilibrium with its surroundings. This prevents unbalanced forces and acceleration of moving system boundaries, which in turn avoids friction and other dissipation.

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