Master equation

In physics, chemistry and related fields, master equations are used to describe the time evolution of a system that can be modelled as being in a probabilistic combination of states at any given time and the switching between states is determined by a transition rate matrix. The equations are a set of differential equations over time of the probabilities that the system occupies each of the different states. In physics, chemistry and related fields, master equations are used to describe the time evolution of a system that can be modelled as being in a probabilistic combination of states at any given time and the switching between states is determined by a transition rate matrix. The equations are a set of differential equations over time of the probabilities that the system occupies each of the different states. A master equation is a phenomenological set of first-order differential equations describing the time evolution of (usually) the probability of a system to occupy each one of a discrete set of states with regard to a continuous time variable t. The most familiar form of a master equation is a matrix form: where P → {displaystyle {vec {P}}} is a column vector (where element i represents state i), and A {displaystyle mathbf {A} } is the matrix of connections. The way connections among states are made determines the dimension of the problem; it is either When the connections are time-independent rate constants, the master equation represents a kinetic scheme, and the process is Markovian (any jumping time probability density function for state i is an exponential, with a rate equal to the value of the connection). When the connections depend on the actual time (i.e. matrix A {displaystyle mathbf {A} } depends on the time, A → A ( t ) {displaystyle mathbf {A} ightarrow mathbf {A} (t)} ), the process is not stationary and the master equation reads When the connections represent multi exponential jumping time probability density functions, the process is semi-Markovian, and the equation of motion is an integro-differential equation termed the generalized master equation: The matrix A {displaystyle mathbf {A} } can also represent birth and death, meaning that probability is injected (birth) or taken from (death) the system, where then, the process is not in equilibrium. Let A {displaystyle mathbf {A} } be the matrix describing the transition rates (also known as kinetic rates or reaction rates). As always, the first subscript represents the row, the second subscript the column. That is, the source is given by the second subscript, and the destination by the first subscript. This is the opposite of what one might expect, but it is technically convenient. For each state k, the increase in occupation probability depends on the contribution from all other states to k, and is given by: where P ℓ {displaystyle P_{ell }} is the probability for the system to be in the state ℓ {displaystyle ell } , while the matrix A {displaystyle mathbf {A} } is filled with a grid of transition-rate constants. Similarly, P k {displaystyle P_{k}} contributes to the occupation of all other states P ℓ , {displaystyle P_{ell },}