Editing Detailed balance
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:<math>\pi(s') P(s',s) = \pi(s) P(s,s')\,.</math> | :<math>\pi(s') P(s',s) = \pi(s) P(s,s')\,.</math> | ||
A Markov process that has detailed balance is said to be a ''reversible Markov process'' or ''reversible Markov chain'' <ref name=OHagan /> | A Markov process that has detailed balance is said to be a '''reversible Markov process''' or '''reversible Markov chain'''.<ref name=OHagan /> | ||
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. Detailed balance implies that, around any closed cycle of states, there is no net flow of probability. For example, it implies that, for all ''a'', ''b'' and ''c'', | 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. Detailed balance implies that, around any closed cycle of states, there is no net flow of probability. For example, it implies that, for all ''a'', ''b'' and ''c'', | ||
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The inequality holds because the logarithm function is monotonic, hence, the expressions <math>\ln w_r^+-\ln w_r^-</math> and <math>w_r^+-w_r^-</math> always have the same sign. | The inequality holds because the logarithm function is monotonic, hence, the expressions <math>\ln w_r^+-\ln w_r^-</math> and <math>w_r^+-w_r^-</math> always have the same sign. | ||
Similar inequalities <ref name=Yab1991/> are valid for other classical conditions for the closed systems and the corresponding characteristic functions: for [[Isothermal-isobaric ensemble |isothermal isobaric conditions]] the [[Gibbs energy function]] decreases, for [[Microcanonical ensemble |isochoric systems with constant internal energy]] the entropy increases as well as for | Similar inequalities <ref name=Yab1991/> are valid for other classical conditions for the closed systems and the corresponding characteristic functions: for [[Isothermal-isobaric ensemble |isothermal isobaric conditions]] the [[Gibbs energy function]] decreases, for [[Microcanonical ensemble |isochoric systems with constant internal energy]] the entropy increases as well as for isobaric systems with the constant [[enthalpy]]. | ||
== Onsager reciprocal relations and detailed balance == | == Onsager reciprocal relations and detailed balance == | ||
Let the principle of detailed balance be valid. Then, in the linear approximation near equilibrium the reaction rates for the generalized mass action law are | Let the principle of detailed balance be valid. Then, in the linear approximation near equilibrium the reaction rates for the generalized mass action law are | ||
:<math>w^+_r=w^{\rm eq}_r \left(1+\sum_i \frac{\alpha_{ri}(\mu_i-\mu^{\rm eq}_i)}{RT}\right); \;\; w^-_r=w^{\rm eq}_r \left(1+ \sum_i \frac{\beta_{ri}(\mu_i-\mu^{\rm eq}_i)}{RT}\right);</math> | :<math>w^+_r=w^{\rm eq}_r \left(1+\sum_i \frac{\alpha_{ri}(\mu_i-\mu^{\rm eq}_i)}{RT}\right); \;\; w^-_r=w^{\rm eq}_r \left(1+ \sum_i \frac{\beta_{ri}(\mu_i-\mu^{\rm eq}_i)}{RT}\right);</math> | ||
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==Semi-detailed balance== | ==Semi-detailed balance== | ||
To formulate the principle of semi-detailed balance, it is convenient to count the direct and inverse elementary reactions separately. In this case, the kinetic equations have the form: | To formulate the principle of semi-detailed balance, it is convenient to count the direct and inverse elementary reactions separately. In this case, the kinetic equations have the form: | ||