77.56.53.183 at 11:03, 27 August 2014

2014-08-27T11:03:08Z

en>Dirac66: 1.Standard notation for units, without double / 2.Remove "so that ...so that"

2013-05-22T17:38:46Z

1.Standard notation for units, without double / 2.Remove "so that ...so that"

← Older revision		Revision as of 19:38, 22 May 2013
Line 1:		Line 1:
	~~Legal Executive Takacs from Comber~~, ~~spends~~ time with ~~passions~~ for ~~example shopping~~, ~~diet~~ and ~~creating dollhouses~~. ~~Intends~~ to ~~retire~~ and ~~take~~ the ~~family~~ to ~~numerous~~ great ~~heritage listed locales on~~ the ~~globe like Pitons Management Area~~.		{{about\|\|the master equation used in quantum physics\|Lindblad equation\|the classical and quantum master equations in quantum field theory\|Batalin–Vilkovisky formalism}}

			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 [[Classical mechanics\|states]] at any given time, and where switching between states is treated [[probability\|probabilistically]]. The equations are usually a set of [[differential equations]] for the variation over time of the [[probability\|probabilities]] that the system occupies each different states.

			==Introduction==
			A master equation is a [[phenomenology (science)\|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 (mathematics)\|set]] of [[Classical mechanics\|states]] with regard to a continuous time variable ''t''. The most familiar form of a master equation is a matrix form:
			:<math> \frac{d\vec{P}}{dt}=\mathbf{A}\vec{P},</math>
			where <math>\vec{P}</math> is a column vector (where element ''i'' represents state ''i''), and <math>\mathbf{A}</math> 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 <math>\mathbf{A}</math> depends on the time, <math>\mathbf{A}\rightarrow\mathbf{A}(t)</math> ), the process is not stationary and the master equation reads
			:<math> \frac{d\vec{P}}{dt}=\mathbf{A}(t)\vec{P}.</math>

			When the connections represent multi exponential [[jumping time]] [[probability density function]]s, 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_\ell, </math> is the probability for the system to be in the state ''<math> \ell </math>'', 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 ([[Microscopic reversibility]]) 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 special case of the master equation is the [[Fokker-Planck equation]] which describes the time evolution of a continuous probability distribution{{Citation needed\|reason=many books, as Van Kampen, introduce the Master Equation for the evolution of a continuos distribution with jumps, while the Fokker-Planck represents diffusion without jumps\|date=January 2014}}. 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]].

			==See also==
			* [[Markov process]]
			* [[Fermi's golden rule]]
			* [[Detailed balance]]
			* [[Boltzmann's H-theorem]]
			* [[Continuous-time Markov process]]

			{{No footnotes\|date=June 2011}}

			==References==
			*{{cite book \| author=van Kampen, N. G. \| title=Stochastic processes in physics and chemistry \| publisher=North Holland \|year=1981 \|isbn=0-444-52965-9 \|isbn=978-0-444-52965-7}}
			*{{cite book \| author=Gardiner, C. W. \| title=Handbook of Stochastic Methods \| publisher=Springer \|year=1985 \|isbn=3-540-20882-8}}
			*{{cite book \| author=Risken, H. \| title=The Fokker-Planck Equation \| publisher=Springer \|year=1984 \|isbn=3-540-61530-X }}

			==External links==
			* Timothy Jones, ''[http://www.physics.drexel.edu/~tim/open/mas/mas.html A Quantum Optics Derivation]'' (2006)

			[[Category:Statistical mechanics]]
			[[Category:Stochastic processes]]
			[[Category:Equations]]
			[[Category:Concepts in physics]]

en>BattyBot: changed {{Unreferenced}} to {{Refimprove}} & general fixes using AWB (8097)

2012-07-14T16:12:53Z

changed {{Unreferenced}} to {{Refimprove}} & general fixes using AWB (8097)

New page

Legal Executive Takacs from Comber, spends time with passions for example shopping, diet and creating dollhouses. Intends to retire and take the family to numerous great heritage listed locales on the globe like Pitons Management Area.

Entropy of fusion - Revision history

77.56.53.183 at 11:03, 27 August 2014

en>Dirac66: 1.Standard notation for units, without double / 2.Remove "so that ...so that"

en>BattyBot: changed {{Unreferenced}} to {{Refimprove}} & general fixes using AWB (8097)