# reversible markov chain

## reversible markov chain

In these cases, a uniform distribution over the states is an equilibrium distribution. , {\displaystyle \mu _{i}=\partial F(T,V,N)/\partial N_{i}} β Pij = P(Xt = j | Xt − 1 = i), and πi and πj are the equilibrium probabilities of being in states i and j, respectively. Here, the reverse reactions with positive constants are included in the list separately. α ) {\displaystyle k_{r}^{-}} [1] The arguments in favor of this property are founded upon microscopic reversibility.[2]. w ∂ . The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). A ∈ 95 (2), 153–164. ln ), detailed mechanisms include both reversible and irreversible reactions. Y For the Markov kinetics the semi-detailed balance condition is just the elementary balance equation and holds for any steady state. ( [8][9][10], The microscopic "reversing of time" turns at the kinetic level into the "reversing of arrows": the elementary processes transform into their reverse processes. 2 + To describe dynamics of the systems that obey the generalized mass action law, one has to represent the activities as functions of the concentrations cj and temperature. is convex because , let us define two sets of numbers: r do not depend on (positive) values of equilibrium reaction rates / [13] When Pr(Xt−1 = i) = πi for all i, this is equivalent to the joint probability matrix, Pr(Xt−1 = i, Xt = j) being symmetric in i and j; or symmetric in t − 1 and t. The definition carries over straightforwardly to continuous variables, where π becomes a probability density, and P(s′, s) a transition kernel probability density from state s′ to state s: The detailed balance condition is stronger than that required merely for a stationary distribution; that is, there are Markov processes with stationary distributions that do not have detailed balance. ∂ [24] Under these microscopic assumptions, the semi-detailed balance condition is just the balance equation for the Markov microkinetics according to the Michaelis–Menten–Stueckelberg theorem. 1 0 A It is sufficient to use in the Wegscheider conditions a basis of solutions of the system w i Thus, the principle of detailed balance is a sufficient but not necessary condition for entropy increase in Boltzmann kinetics. is symmetric: These symmetry relations, Transition matrices that are symmetric (Pij = Pji or P(s′, s) = P(s, s′)) always have detailed balance. 95 (2), 115–152. μ where Lorentz H.-A. m ⟶ Now, after almost 150 years of development, the scope of validity and the violations of detailed balance in kinetics seem to be clear. + = Wegscheider's conditions for the generalized mass action law, Dissipation in systems with detailed balance, Onsager reciprocal relations and detailed balance, Dissipation in systems with semi-detailed balance, Cone theorem and local equivalence of detailed and complex balance, Detailed balance for systems with irreversible reactions. = The Boltzmann formula (1872) for entropy production in rarefied gas kinetics with detailed balance[1][2] served as a prototype of many similar formulas for dissipation in mass action kinetics[14] and generalized mass action kinetics[15] with detailed balance. c is the reaction rate constant for the reverse reaction if it is in the reaction mechanism and 0 if it is not; ( + Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien. ): This is exactly the Onsager form: following the original work of Onsager,[6] we should introduce the thermodynamic forces = + A ~ {\displaystyle w_{r}^{+}-w_{r}^{-}} ⟶ φ T ρ = N ( ( ′ v D is the rth row of {\displaystyle \gamma _{r}} This means that the system of linear detailed balance equations. V r ) j ⁡

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