Master equation

A master equation is a phenomenologically founded differential equation of the first order that describes the time evolution of the probabilities of a system.

description

For states from a discrete set of states the master equation is:

${\displaystyle {\frac {\mathrm {d} P_{k}}{\mathrm {d} t}}=\sum _{\ell \neq k}(T_{k\ell }P_{\ell }-T_{\ell k}P_{k}).}$

where is the probability that the system is in the state and is the assumed constant transition probability rate from state to state . The master equation for continuous states (and corresponding probability densities) can be formulated analogously, only with an integration instead of a summation as in the case of discrete states. ${\displaystyle P_{k}}$ ${\displaystyle k}$ ${\displaystyle T_{k\ell }}$ ${\displaystyle \ell }$ ${\displaystyle k}$

In probability theory , this is considered a continuous Markov process in which the integrated master equation corresponds to the Chapman-Kolmogorow equation .

If the matrix is symmetric (i.e. all microscopic transitions are reversible and the transition probability rates are the same in both directions), then: ${\displaystyle T_{\ell k}}$

${\displaystyle T_{k\ell }=T_{\ell k},}$

and thus:

${\displaystyle {\frac {\mathrm {d} P_{k}}{\mathrm {d} t}}=\sum _{\ell }T_{k\ell }(P_{\ell }-P_{k}).}$

The master equation (an integro differential equation ) can be expressed as a partial differential equation of infinite order: one then speaks of the Kramers-Moyal expansion .

application

The master equation can be used to describe the time evolution of a statistical observable : where the master equation can be used in the back part. This can (after the introduction of the jump moments) be used to derive the linear response theory . ${\displaystyle x}$ ${\displaystyle E(x)(t)=\sum _{x}xP_{x}(t)\implies {\frac {dE(x)}{dt}}(t)=\sum _{x}x{\frac {dP_{x}}{dt}}(t)}$

The master equation in the above form was first derived from the quantum statistics by Wolfgang Pauli and is therefore also called the Pauli master equation. It is a differential equation for the state probabilities, i.e. the diagonal elements of the density matrix . There are also generalizations that include the off-diagonal elements ( master equation in Lindblad form ). Another generalization is the Nakajima twenty equation in the Mori twenty formalism .

More generally, in statistical mechanics, master equations are called basic equations (often in the form of a balance equation above) for the probability distributions, from which equations that are easier to solve can then be derived through approximations and limit crossings, such as differential equations of the type of the Fokker-Planck equation ( which also includes the diffusion equation) in the continuum limit. However, behind these approximations there is still the microscopically valid master equation , hence the name.

literature

• Hartmut Haug: Statistical Physics - Equilibrium Theory and Kinetics . 2nd Edition. Springer 2006, ISBN 3-540-25629-6 .

See also

• Markov trial

Individual evidence

1. ^ Van Kampen Stochastic problems in physics and chemistry , North Holland, Chapter V, Master Equation
2. ↑ Stochastic Processes: From Physics to Finance, Paul, Baschnagel, p. 47
3. ↑ z. B. AJ Fisher Lectures on open quantum systems 2004 ( Memento of the original from May 23, 2009 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.