Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Master equation
Summary
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.
When the probabilities of the elementary processes are known, one can write down a continuity equation for W, from which all other equations can be derived and which we will call therefore the "master” equation.
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 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
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.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.