Master equation
Encyclopedia
In physics
and chemistry
and related fields, master equations are used to describe the time-evolution of a system that can be modelled as being in exactly one of countable number of states at any given time, and where switching between states is treated probabilistically
. The equations are usually a set of differential equations for the variation over time of the probabilities
that the system occupies each different states.
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
is a column vector (where element i represents state i), and 
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
depends on the time, 
), the process is not Markovian, and the master equation obeys,
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
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.
Detailed description of the matrix
Let 
be the matrix describing the transition rates (also known, kinetic rates or reaction rates). The element 
is the rate constant that corresponds to the transition from state k to state ℓ. Since 
is square, the indices ℓ and k may be arbitrarily defined as rows or columns. Here, the first subscript is row, the second is column. The order of the subscripts, which refer to source and destination states, are opposite of the normal convention for elements of a matrix. That is, in other contexts, 
could be interpreted as the 
transition. However, it is convenient to write the subscripts in the opposite order when using Einstein notation
, so the subscripts in
should be interpreted as 
.
For each state k, the increase in occupation probability depends on the contribution from all other states to k, and is given by:
where
is the probability for the system to be in the state k, while the matrix
is filled with a grid of transition-rate constant
s. Similarly, Pk contributes to the occupation of all other states:
:
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 the
is not defined or has been assigned an arbitrary value.
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 
,
These symmetry relations were proved on the basis of the time reversibility
of microscopic dynamics (as Onsager reciprocal relations
).
, 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 generalization of the master equation is the Fokker–Planck equation which describes the time evolution of a continuous probability distribution.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
and related fields, master equations are used to describe the time-evolution of a system that can be modelled as being in exactly one of countable number of states at any given time, and where switching between states is treated probabilistically
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
. The equations are usually a set of differential equations for the variation over time of the probabilities
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
that the system occupies each different states.
Introduction
A master equation is a phenomenologicalPhenomenology (science)
The term phenomenology in science is used to describe a body of knowledge that relates empirical observations of phenomena to each other, in a way that is consistent with fundamental theory, but is not directly derived from theory. For example, we find the following definition in the Concise...
set of first-order differential equations describing the time evolution of (usually) the probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
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
is a column vector (where element i represents state i), 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
Kinetic scheme
In physics and chemistry and related fields, a kinetic scheme is a network of states and connections among the states representing the scheme of a dynamical process. Usually, a kinetic scheme represents a Markovian process, where when the process is not Markovian, the scheme is a generalized...
, 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 Markovian, and the master equation obeys,When the connections represent multi exponential jumping time probability density functions, the process is semi-Markovian
Semi-Markov process
A continuous-time stochastic process is called a semi-Markov process or 'Markov renewal process' if the embedded jump chain is a Markov chain, and where the holding times are random variables with any distribution, whose distribution function may depend on the two states between which the move is...
, and the equation of motion is an integro-differential equation
Integro-differential equation
An integro-differential equation is an equation which involves both integrals and derivatives of a function.The general first-order, linear integro-differential equation is of the form...
termed the generalized master equation:
The matrix
can also represent birth and deathBirth-death process
The birth–death process is a special case of continuous-time Markov process where the states represent the current size of a population and where the transitions are limited to births and deaths...
, meaning that probability is injected (birth) or taken from (death) the system, where 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, kinetic rates or reaction rates). The element is the rate constant that corresponds to the transition from state k to state ℓ. Since is square, the indices ℓ and k may be arbitrarily defined as rows or columns. Here, the first subscript is row, the second is column. The order of the subscripts, which refer to source and destination states, are opposite of the normal convention for elements of a matrix. That is, in other contexts, could be interpreted as the transition. However, it is convenient to write the subscripts in the opposite order when using Einstein notationEinstein notation
In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...
, so the subscripts in
should be interpreted as .For each state k, the increase in occupation probability depends on the contribution from all other states to k, and is given by:
where
is the probability for the system to be in the state k, while the matrixMatrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
is filled with a grid of transition-rate constantConstant (mathematics)
In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition...
s. Similarly, Pk contributes to the occupation of all other states:
: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 the
is not defined or has been assigned an arbitrary value.The master equation exhibits detailed balance
Detailed balance
The principle of detailed balance is formulated for kinetic systems which are decomposed into elementary processes : At equilibrium, each elementary process should be equilibrated by its reverse process....
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 ,These symmetry relations were proved on the basis of the time reversibility
Time reversibility
Time reversibility is an attribute of some stochastic processes and some deterministic processes.If a stochastic process is time reversible, then it is not possible to determine, given the states at a number of points in time after running the stochastic process, which state came first and which...
of microscopic dynamics (as Onsager reciprocal relations
Onsager reciprocal relations
In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists....
).
Examples of master equations
Many physical problems in classicalClassical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
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
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
).
The Lindblad equation
Lindblad equation
In quantum mechanics, the Kossakowski–Lindblad equation or master equation in the Lindblad form is the most general type of markovian and time-homogeneous master equation describing non-unitary evolution of the density matrix \rho that is trace preserving and completely positive for any initial...
in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
is a generalization of the master equation describing the time evolution of a density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
. 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 generalization of the master equation is the Fokker–Planck equation which describes the time evolution of a continuous probability distribution.
See also
- Markov processMarkov processIn probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...
- Fermi's golden ruleFermi's golden ruleIn quantum physics, Fermi's golden rule is a way to calculate the transition rate from one energy eigenstate of a quantum system into a continuum of energy eigenstates, due to a perturbation....
- Detailed balanceDetailed balanceThe principle of detailed balance is formulated for kinetic systems which are decomposed into elementary processes : At equilibrium, each elementary process should be equilibrated by its reverse process....
- Boltzmann's H-theorem
- Continuous-time Markov process
External links
- Timothy Jones, A Quantum Optics Derivation (2006)