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{{Dablink|See [[Lindblad equation]] for the master equation used in quantum physics}}
{{Dablink|See also [[Batalin–Vilkovisky formalism]] for the classical and quantum master equations in quantum field theory.}}
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Em [[física]] e [[química]] e campos relacionados, '''equações mestre''' são usadas para descrever a evolução no tempo de um sistema que pode ser modelado como estando em um exato número contável de [[estado (física)|estados]] a qualquer tempo dado, e onde a divisão entre estados é tratada [[probabilidade|probabilisticamente]]. As equações são usualmente um conjunto de [[equação diferencial|equações diferenciais]] para a variação no tempo das [[probabilidade]]s que tal sistema ocupa em cada diferente estado.
*uma rede, onde cada par de estados pode ter uma conexão (dependendo das propriedades da rede).
Quando as conexões são simplesmente números<!-- {{Clarify|date=June 2011}} -->, a equação mestre representa um [[esquema cinético]], e o processo é [[Markoviano]] (qualquer salto de tempo da função densidade de probabilidade para o estado ''i'' é um exponencial, com uma taxa igual ao valor da conexão). Quando as conexões dependem do tempo atual (''i.e.'' a matriz <math>\mathbf{A}</math> depende do tempo, <math>\mathbf{A}\rightarrow\mathbf{A}(t)</math> ), o processo não é Markoviano, e a equação mestre obedece,
:<math> \frac{d\vec{P}}{dt}=\mathbf{A}(t)\vec{P}.</math>
{{referências}}
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== Bibliografia ==
When the connections represent multi exponential [[jumping time]] [[probability density functions]], the process is [[Semi-Markov process|semi-Markovian]], and the equation of motion is an [[integro-differential equation]] termed the generalized master equation:
:<math> \frac{d\vec{P}}{dt}= \int^t_0 \mathbf{A}(t- \tau )\vec{P}( \tau )d \tau . </math>
The matrix <math>\mathbf{A}</math> can also represent [[Birth-death process|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 <math>\mathbf{A}</math>, and properties of the system===
Let <math>\mathbf{A}</math> be the matrix describing the transition rates (also known, kinetic rates or reaction rates). The element <math>\scriptstyle A_{\ell k}</math> is the rate constant that corresponds to the transition from state ''k'' to state ℓ. Since <math>\mathbf{A}</math> 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, <math>A_{12}</math> could be interpreted as the <math>1 \rightarrow 2</math> transition. However, it is convenient to write the subscripts in the opposite order when using [[Einstein notation]], so the subscripts in <math>A_{12}</math> should be interpreted as <math>1 \leftarrow 2</math>.
For each state ''k'', the increase in occupation probability depends on the contribution from all other states to ''k'', and is given by:
:<math> \sum_\ell A_{k\ell}P_\ell, </math>
where <math> P_k, </math> is the probability for the system to be in the state ''k'', while the [[matrix (mathematics)|matrix]] <math>\mathbf{A}</math> is filled with a grid of transition-rate [[Constant (mathematics)|constant]]s. Similarly, ''P<sub>k</sub>'' contributes to the occupation of all other states:<math> P_\ell, </math>:
:<math> \sum_\ell A_{\ell k}P_k, </math>
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 <math>\mathbf{A}</math> is not defined or has been assigned an arbitrary value.
:<math> \frac{dP_k}{dt}=\sum_\ell(A_{k\ell}P_\ell - A_{\ell k}P_k)=\sum_{\ell\neq k}(A_{k\ell}P_\ell - A_{\ell k}P_k). </math>
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 <math>\scriptstyle\pi_k</math> and <math>\scriptstyle\pi_\ell</math>,
:<math>A_{k \ell} \pi_\ell = A_{\ell k} \pi_k .</math>
These symmetry relations were proved on the basis of the [[time reversibility]] of microscopic dynamics (as [[Onsager reciprocal relations]]).
=== Examples of master equations===
Many physical problems in [[classical mechanics|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 generalization of the master equation is the [[Fokker–Planck equation]] which describes the time evolution of a continuous probability distribution.
==See also==
* [[Markov process]]
* [[Fermi's golden rule]]
* [[Detailed balance]]
* [[Boltzmann's H-theorem]]
{{No footnotes|date=June 2011}}
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{{em tradução|:en:Master equation}}
==Referências==
<references/>
*{{citar livro|autor =van Kampen, N. G. |título=Stochastic processes in physics and chemistry |publicado=North Holland |ano=1981 |isbn=978-0-444-52965-7}}
*{{citar livro|autor =Gardiner, C. W. |título=Handbook of Stochastic Methods |publicado=Springer |ano=1985 |isbn=3-540-20882-8}}
*{{citar livro|autor =Risken, H. |título=The Fokker-Planck Equation |publicado=Springer |ano=1984 |isbn=3-450-61530-X}}
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==External links==
* Timothy Jones, ''[http://www.physics.drexel.edu/~tim/open/mas/mas.html A Quantum Optics Derivation]'' (2006)
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[[Categoria:Mecânica estatística]]
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