Master equation

In physics, chemistry, and related fields, master equations are used to describe the time evolution of a system that can be modeled 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.

Etymology

The name was proposed in 1940:

Formulation

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: \frac{d\vec{P}}{dt} = \mathbf{A}\vec{P}, where \vec{P} is a column vector, and \mathbf{A} is the matrix of connections. The way connections among states are made determines the dimension of the problem; it is either a d-dimensional system (where d is 1,2,3,...), where any state is connected with exactly its 2d nearest neighbors, or a network, where every pair of states may have a connection (depending on the network's properties). 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 \mathbf{A} depends on the time, \mathbf{A}\rightarrow\mathbf{A}(t) ), the process is not stationary and the master equation reads \frac{d\vec{P}}{dt} = \mathbf{A}(t)\vec{P}. 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: \frac{d\vec{P}}{dt}= \int^t_0 \mathbf{A}(t- \tau )\vec{P}( \tau ) \, d \tau . The transition rate matrix \mathbf{A} can also represent birth and death, meaning that probability is injected (birth) or taken from (death) the system, and then the process is not in equilibrium. When the transition rate matrix can be related to the probabilities, one obtains the Kolmogorov equations.

Detailed description of the matrix and properties of the system

Let \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 is appropriate for conventional matrix multiplication. For each state k, the increase in occupation probability depends on the contribution from all other states to k, and is given by: \sum_\ell A_{k\ell}P_\ell, where P_\ell is the probability for the system to be in the state \ell , while the matrix \mathbf{A} is filled with a grid of transition-rate constants. Similarly, P_k contributes to the occupation of all other states P_\ell, \sum_\ell A_{\ell k}P_k, In probability theory, this identifies the evolution as a continuous-time Markov process, with the integrated master equation obeying a Chapman–Kolmogorov equation. The master equation can be simplified so that the terms with \ell=k do not appear in the summation. This allows calculations even if the main diagonal of \mathbf{A} is not defined or has been assigned an arbitrary value. \frac{dP_k}{dt} = \sum_{\ell\neq k}(A_{k\ell}P_\ell - A_{\ell k}P_k). The final equality arises from the fact that \sum_{\ell, k}(A_{\ell k}P_k) = \frac{d}{dt} \sum_\ell(P_{\ell}) = 0 because the summation over the probabilities P_{\ell} yields one, a constant function. Since this has to hold for any probability \vec{P} (and in particular for any probability of the form P_{\ell} = \delta_{\ell k} for some k) we get \sum_{\ell}(A_{\ell k}) = 0 \qquad \forall k. Using this we can write the diagonal elements as A_{kk} = -\sum_{\ell\neq k}(A_{\ell k}) \Rightarrow A_{kk} P_k = -\sum_{\ell\neq k}(A_{\ell k} P_k) . The master equation exhibits detailed balance if each of the terms of the summation disappears separately at equilibrium—i.e. if, for all states k and ℓ having equilibrium probabilities \pi_k and \pi_\ell , A_{k \ell} \pi_\ell = A_{\ell k} \pi_k . These symmetry relations were proved on the basis of the time reversibility of microscopic dynamics (microscopic reversibility) as Onsager reciprocal relations.

Examples of master equations

Many physical problems in classical, quantum mechanics and problems in other sciences, can be reduced to the form of a master equation, thereby performing a great simplification of the problem (see mathematical model). The Lindblad equation in quantum mechanics is a generalization of the master equation describing the time evolution of a density matrix. Though the Lindblad equation is often referred to as a master equation, it is not one in the usual sense, as it governs not only the time evolution of probabilities (diagonal elements of the density matrix), but also of variables containing information about quantum coherence between the states of the system (non-diagonal elements of the density matrix). Another special case of the master equation is the Fokker–Planck equation which describes the time evolution of a continuous probability distribution. Complicated master equations which resist analytic treatment can be cast into this form (under various approximations), by using approximation techniques such as the system size expansion. Stochastic chemical kinetics provide yet another example of the use of the master equation. A master equation may be used to model a set of chemical reactions when the number of molecules of one or more species is small (of the order of 100 or 1000 molecules). The chemical master equation can also solved for the very large models, such as the DNA damage signal from fungal pathogen Candida albicans. \sum_{\ell}A_{\ell k} = 0 \qquad \forall k and A_{\ell k} \geq 0 \qquad \forall \ell\neq k, one can show using the Perron–Frobenius theorem that: There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of \mathbf{A} is strongly connected. All other eigenvalues \lambda fulfill 0 > \operatorname{Re} \lambda \geq 2 \operatorname{min}_i A_{ii} . All eigenvectors v with a non-zero eigenvalue fulfill \sum_{i}v_{i} = 0 . This has important consequences for the time evolution of a state.

References

External links

Timothy Jones, A Quantum Optics Derivation (2006)

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