Master equation explained

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.

The name was proposed in 1940:^{[1] [2]}

Introduction

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:$$\backslash frac\; =\; \backslash mathbf\backslash vec,$$where

is a column vector, and 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

depends on the time, ), the process is not stationary and the master equation reads$$\backslash frac\; =\; \backslash mathbf(t)\backslash vec.$$

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:$$\backslash frac=\; \backslash int^t\_0\; \backslash mathbf(t-\; \backslash tau)\backslash vec(\backslash tau)\; \backslash ,\; d\; \backslash tau\; .$$

The matrix

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.

Detailed description of the matrix and properties of the system

Let

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:$$\backslash sum\_\backslash ell\; A\_P\_\backslash ell,$$where

is the probability for the system to be in the state , while the matrix is filled with a grid of transition-rate constants. Similarly, contributes to the occupation of all other states $$\backslash sum\_\backslash ell\; A\_P\_k,$$

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 ℓ = k do not appear in the summation. This allows calculations even if the main diagonal of

is not defined or has been assigned an arbitrary value.$$\backslash frac\; =\; \backslash sum\_\backslash ell(A\_P\_\backslash ell)\; =\; \backslash sum\_(A\_P\_\backslash ell)\; +\; A\_P\_k\; =\; \backslash sum\_(A\_P\_\backslash ell\; -\; A\_P\_k).$$

The final equality arises from the fact that $$\backslash sum\_(A\_P\_k)\; =\; \backslash frac\; \backslash sum\_\backslash ell(P\_)\; =\; 0$$ because the summation over the probabilities

yields one, a constant function. Since this has to hold for any probability (and in particular for any probability of the form for some k) we get$$\backslash sum\_(A\_)\; =\; 0\; \backslash qquad\; \backslash forall\; k.$$ Using this we can write the diagonal elements as$$A\_\; =\; -\backslash sum\_(A\_)\; \backslash Rightarrow\; A\_\; P\_k\; =\; -\backslash sum\_(A\_\; P\_k)\; .$$

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

and ,$$A\_\; \backslash pi\_\backslash ell\; =\; A\_\; \backslash pi\_k\; .$$

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.^[3] 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).^[4] The chemical master equation can also solved for the very large models, such as the DNA damage signal from fungal pathogen Candida albicans.^[5]

Quantum master equations

A quantum master equation is a generalization of the idea of a master equation. Rather than just a system of differential equations for a set of probabilities (which only constitutes the diagonal elements of a density matrix), quantum master equations are differential equations for the entire density matrix, including off-diagonal elements. A density matrix with only diagonal elements can be modeled as a classical random process, therefore such an "ordinary" master equation is considered classical. Off-diagonal elements represent quantum coherence which is a physical characteristic that is intrinsically quantum mechanical.

The Redfield equation and Lindblad equation are examples of approximate quantum master equations assumed to be Markovian. More accurate quantum master equations for certain applications include the polaron transformed quantum master equation, and the VPQME (variational polaron transformed quantum master equation).^[6]

Theorem about eigenvalues of the matrix and time evolution

Because

fulfills$$\backslash sum\_A\_\; =\; 0\; \backslash qquad\; \backslash forall\; k$$and $$A\_\; \backslash geq\; 0\; \backslash qquad\; \backslash forall\; \backslash ell\backslash neq\; k,$$ one can show^[7] that:

• There is at least one eigenvector with a vanishing eigenvalue, exactly one if the graph of

is strongly connected.

• All other eigenvalues

fulfill .

• All eigenvectors

with a non-zero eigenvalue fulfill $\backslash sum\_v\_\; =\; 0$.

This has important consequences for the time evolution of a state.

See also

• Kolmogorov equations (Markov jump process)
• Continuous-time Markov process
• Quantum master equation
• Fermi's golden rule
• Detailed balance
• Boltzmann's H-theorem

References

• Book: van Kampen, N. G. . Stochastic processes in physics and chemistry . registration . North Holland . 1981 . 978-0-444-52965-7.
• Book: Gardiner, C. W. . Handbook of Stochastic Methods . Springer . 1985 . 978-3-540-20882-2.
• Book: Risken, H. . The Fokker-Planck Equation . Springer . 1984 . 978-3-540-61530-9 .

External links

• Timothy Jones, A Quantum Optics Derivation (2006)

Notes and References

1. Cohen . E. G. D. . July 1990 . George E. Uhlenbeck and statistical mechanics . American Journal of Physics . en . 58 . 7 . 619–625 . 10.1119/1.16504 . 1990AmJPh..58..619C . 0002-9505.
2. Nordsieck . A. . Lamb . W.E. . Uhlenbeck . G.E. . 1940 . On the theory of cosmic-ray showers I the furry model and the fluctuation problem . Physica . en . 7 . 4 . 344–360 . 10.1016/S0031-8914(40)90102-1. 1940Phy.....7..344N . 2027.42/32597 . free .
3. Book: Honerkamp. Josef. Statistical physics : an advanced approach with applications; with 7 tables and 57 problems with solutions. limited. 1998. Springer. Berlin [u.a.]. 978-3-540-63978-7 . 173.
4. Gupta. Ankur. Rawlings. James B.. Apr 2014. Comparison of Parameter Estimation Methods in Stochastic Chemical Kinetic Models: Examples in Systems Biology. AIChE Journal. 60. 4. 1253–1268. 10.1002/aic.14409. 0001-1541. 4946376. 27429455. 2014AIChE..60.1253G .
5. Kosarwal. Rahul. Kulasiri. Don. Samarasinghe. Sandhya. Nov 2020. Novel domain expansion methods to improve the computational efficiency of the Chemical Master Equation solution for large biological networks. BMC Bioinformatics . 21 . 1 . 515 . 10.1186/s12859-020-03668-2 . 33176690 . 7656229 . free .
6. McCutcheon . D. . Dattani . N. S. . Gauger . E. . Lovett . B. . Nazir . A. . 25 August 2011 . A general approach to quantum dynamics using a variational master equation: Application to phonon-damped Rabi rotations in quantum dots . Physical Review B . 84 . 8 . 081305R . 10.1103/PhysRevB.84.081305 . 1105.6015 . 2011PhRvB..84h1305M. 10044/1/12822 . 119275166 .
7. Keizer. Joel. 1972-11-01. On the solutions and the steady states of a master equation. Journal of Statistical Physics. en. 6. 2. 67–72. 10.1007/BF01023679. 1972JSP.....6...67K . 120377514 . 1572-9613.