Detailed balance - Revision history

Carl McBride: Added book ISBN's

2012-02-08T16:01:01Z

Added book ISBN's

← Older revision		Revision as of 18:01, 8 February 2012
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	In 1901, R. Wegscheider introduced the principle of detailed balance for chemical kinetics.<ref>[http://dx.doi.org/10.1007/BF01517498 Rud Wegscheider "Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme", Monatshefte für Chemie '''22''' pp. 849-906 (1901)]</ref> In particular, he demonstrated that the irreversible cycles <math>A_1 \to A_2 \to ... \to A_n \to A_1</math> are impossible and found explicitly the relations between kinetic constants that follow from the principle of detailed balance.		In 1901, R. Wegscheider introduced the principle of detailed balance for chemical kinetics.<ref>[http://dx.doi.org/10.1007/BF01517498 Rud Wegscheider "Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme", Monatshefte für Chemie '''22''' pp. 849-906 (1901)]</ref> In particular, he demonstrated that the irreversible cycles <math>A_1 \to A_2 \to ... \to A_n \to A_1</math> are impossible and found explicitly the relations between kinetic constants that follow from the principle of detailed balance.

	~~See also <ref name=vanKampen1992>van Kampen, N.G. "Stochastic Processes in Physics and Chemistry", Elsevier Science (1992).</ref>~~
	~~<ref>Lifshitz, E.M., Pitaevskii, L.P., Physical kinetics (1981)London: Pergamon. Vol. 10 of the Course of Theoretical Physics(3rd Ed).</ref>~~

	==Microscopic background==		==Microscopic background==
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	== Detailed balance for systems with irreversible reactions ==		== Detailed balance for systems with irreversible reactions ==

	Detailed balance states that in equilibrium each elementary process is equilibrated by its reverse process and required reversibility of all elementary processes. For many real physico-chemical complex systems (e.g. homogeneous combustion, heterogeneous catalytic oxidation, most enzyme reactions etc), detailed mechanisms include both reversible and irreversible reactions. If one represents irreversible reactions as limits of reversible steps, then it become obvious that not all reaction mechanisms with irreversible reactions can be obtained as limits of systems or reversible reactions with detailed balance. For example, the irreversible cycle <math>A_1 \to A_2 \to A_3 \to A_1</math> cannot be obtained as such a limit but the reaction mechanism <math>A_1 \to A_2 \to A_3 \leftarrow A_1</math> can.<ref>~~Chu, Ch~~. (~~1971~~), Gas absorption accompanied by a system of first-order reactions, ~~Chem. Eng~~. ~~Sci. 26~~(3)~~, 305-312.~~</ref>		Detailed balance states that in equilibrium each elementary process is equilibrated by its reverse process and required reversibility of all elementary processes. For many real physico-chemical complex systems (e.g. homogeneous combustion, heterogeneous catalytic oxidation, most enzyme reactions etc), detailed mechanisms include both reversible and irreversible reactions. If one represents irreversible reactions as limits of reversible steps, then it become obvious that not all reaction mechanisms with irreversible reactions can be obtained as limits of systems or reversible reactions with detailed balance. For example, the irreversible cycle <math>A_1 \to A_2 \to A_3 \to A_1</math> cannot be obtained as such a limit but the reaction mechanism <math>A_1 \to A_2 \to A_3 \leftarrow A_1</math> can <ref>[http://dx.doi.org/10.1016/0009-2509(71)83005-1 Chieh Chu "Gas absorption accompanied by a system of first-order reactions", Chemical Engineering Science '''26''' pp. 305–312 (1971)]</ref>.


	''A system of reactions with some irreversible reactions is a limit of systems with detailed balance when some constants tend to zero if and only if (i) the reversible part of this system satisfies the principle of detailed balance and (ii) the convex hull of the stoichiometric vectors of the irreversible reactions has empty intersection with the linear span of the stoichiometric vectors of the reversible reactions.''<ref name=GorbanYablonsky2011/> Physically, the last condition means that the irreversible reactions cannot be included in oriented cyclic pathways.		''A system of reactions with some irreversible reactions is a limit of systems with detailed balance when some constants tend to zero if and only if (i) the reversible part of this system satisfies the principle of detailed balance and (ii) the convex hull of the stoichiometric vectors of the irreversible reactions has empty intersection with the linear span of the stoichiometric vectors of the reversible reactions.''<ref name=GorbanYablonsky2011/> Physically, the last condition means that the irreversible reactions cannot be included in oriented cyclic pathways.
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	==References==		==References==
	<references/>		<references/>
	'''~~Related reading~~'''		;Related reading
			*[http://dx.doi.org/10.1063/1.477973 Vasilios I. Manousiouthakis and Michael W. Deem "Strict detailed balance is unnecessary in Monte Carlo simulation", Journal of Chemical Physics '''110''' pp. 2753- (1999)]
			*van Kampen, N.G. "Stochastic Processes in Physics and Chemistry", Elsevier Science (1992) ISBN 0444893490
			*E. M. Lifshitz, and L. P. Pitaevskii "Physical kinetics" Butterworth-Heinemann (1981) ISBN 0750626356
			;External links
	*[http://en.wikipedia.org/wiki/Detailed_balance Detailed balance] in Wikipedia		*[http://en.wikipedia.org/wiki/Detailed_balance Detailed balance] in Wikipedia
	*[http://dx.doi.org/10.1063/1.477973 Vasilios I. Manousiouthakis and Michael W. Deem "Strict detailed balance is unnecessary in Monte Carlo simulation", Journal of Chemical Physics '''110''' pp. 2753- (1999)]
	[[category: statistical mechanics]]		[[category: statistical mechanics]]
	[[category: Non-equilibrium thermodynamics]]		[[category: Non-equilibrium thermodynamics]]

Carl McBride: /* Semi-detailed balance */ Slight tidy of references

2012-02-08T15:43:25Z

Semi-detailed balance: Slight tidy of references

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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:

	:<math>\frac{d N_i}{d t}=V\sum_r \gamma_{ri} w_r=V\sum_r (\beta_{ri}-\alpha_{ri})w_r </math>		:<math>\frac{d N_i}{d t}=V\sum_r \gamma_{ri} w_r=V\sum_r (\beta_{ri}-\alpha_{ri})w_r </math>
	Let us use the notations <math>\alpha_r=\alpha_{ri}</math>, <math>\beta_r=\beta_{ri}</math> for the input and the output vectors of the stoichiometric coefficients of the ''r''th elementary reaction. Let <math>Y</math> be the set of all these vectors <math>\alpha_r, \beta_r</math>.

			Let us use the notation <math>\alpha_r=\alpha_{ri}</math>, <math>\beta_r=\beta_{ri}</math> for the input and the output vectors of the stoichiometric coefficients of the ''r''th elementary reaction. Let <math>Y</math> be the set of all these vectors <math>\alpha_r, \beta_r</math>.
	For each <math>\nu \in Y</math>, let us define two sets of numbers:		For each <math>\nu \in Y</math>, let us define two sets of numbers:

	:<math>R_{\nu}^+=\{r\|\alpha_r=\nu \}; \;\;\; R_{\nu}^-=\{r\|\beta_r=\nu \}</math>		:<math>R_{\nu}^+=\{r\|\alpha_r=\nu \}; \;\;\; R_{\nu}^-=\{r\|\beta_r=\nu \}</math>

	<math>r \in R_{\nu}^+</math> if and only if <math>\nu</math> is the vector of the input stoichiometric coefficients <math>\alpha_r</math> for the ''r''th elementary reaction;<math>r \in R_{\nu}^-</math> if and only if <math>\nu</math> is the vector of the output stoichiometric coefficients <math>\beta_r</math> for the ''r''th elementary reaction.		<math>r \in R_{\nu}^+</math> if and only if <math>\nu</math> is the vector of the input stoichiometric coefficients <math>\alpha_r</math> for the ''r''th elementary reaction;<math>r \in R_{\nu}^-</math> if and only if <math>\nu</math> is the vector of the output stoichiometric coefficients <math>\beta_r</math> for the ''r''th elementary reaction.

	The principle of '''semi-detailed balance''~~' means~~ that in equilibrium ~~the semi-detailed balance condition holds:~~ for every <math>\nu \in Y</math>		The principle of ''semi-detailed balance'' implies that when in equilibrium, for every <math>\nu \in Y</math>
	:<math>\sum_{r\in R_{\nu}^+}w_r=\sum_{r\in R_{\nu}^+}w_r</math>		:<math>\sum_{r\in R_{\nu}^+}w_r=\sum_{r\in R_{\nu}^+}w_r</math>

	The semi-detailded balance condition is sufficient for the stationarity: it implies that		The semi-detailded balance condition is sufficient for the stationarity: it implies that
	:<math>\frac{d N}{dt}=V \sum_r \gamma_r w_r=0</math>.		:<math>\frac{d N}{dt}=V \sum_r \gamma_r w_r=0.</math>

	For the Markov kinetics the semi-detailed balance condition is just the elementary balance equation and holds for any steady state. For the nonlinear mass action law it is, in general, sufficient but not necessary condition for stationarity.

	~~The semi-detailed balance condition is weaker than the detailed balance one: if the principle of detailed balance holds then the condition of semi-detailed balance also holds.~~

	For systems that obey the ~~generalized~~ mass action law the semi-detailed balance condition is sufficient for the dissipation inequality <math>d F/ dt \geq 0</math> (for the Helmholtz free energy under isothermal isochoric conditions and for the dissipation inequalities under other classical conditions for the corresponding thermodynamic potentials).		For the Markov kinetics the semi-detailed balance condition is simply the elementary balance equation and holds for any steady state. For the non-linear mass action law it is, in general, a sufficient but not necessary condition for stationarity.
			The semi-detailed balance condition is weaker than that of detailed balance: if the principle of detailed balance holds then the condition of semi-detailed balance also holds.
			For systems that obey the generalised mass action law the semi-detailed balance condition is sufficient for the dissipation inequality <math>d F/ dt \geq 0</math> (for the Helmholtz free energy under isothermal isochoric conditions and for the dissipation inequalities under other classical conditions for the corresponding thermodynamic potentials).

	Boltzmann introduced the semi-detailed balance condition for collisions in 1887<ref name=Boltzmann1887 /> and proved that it guaranties the positivity of the entropy production. For chemical kinetics, this condition (as the ''complex balance'' condition) was ~~inroduced~~ by Horn and Jackson in 1972.<ref name="HornJackson1972">~~''Horn,~~ F.~~, Jackson,~~ R.~~'' (1972)~~ General mass action kinetics. ~~Arch. Ration. Mech. Anal. 47, 87~~-116.</ref>		Boltzmann introduced the semi-detailed balance condition for collisions in 1887 <ref name=Boltzmann1887 /> and proved that it guaranties the positivity of the entropy production. For chemical kinetics, this condition (as the ''complex balance'' condition) was introduced by Horn and Jackson in 1972
			<ref name="HornJackson1972">[http://dx.doi.org/10.1007/BF00251225 F. Horn and R. Jackson "General mass action kinetics", Archive for Rational Mechanics and Analysis '''47''' pp. 81-116 (1972)]</ref>.

	The microscopic backgrounds for the semi-detailed balance were found in the Markov ~~microkinetics~~ of the intermediate compounds that are present in small amounts and whose concentrations are in quasiequilibrium with the main components.<ref>''Stueckelberg~~, E.C.G.'' (1952)~~ Theoreme ''H'' et unitarite de ''S''. ~~Helv. Phys. Acta 25,~~ 577-580</ref> 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.<ref name="GorbanShahzad2011">~~''Gorban, A.N., Shahzad, M.'' (2011)~~ [http://~~arxiv~~.org/~~pdf~~/~~1008~~.~~3296v3~~ The Michaelis-Menten-Stueckelberg Theorem.] Entropy 13~~, no~~. 5, 966-1019.</ref>		The microscopic backgrounds for the semi-detailed balance were found in the Markov micro-kinetics of the intermediate compounds that are present in small amounts and whose concentrations are in quasiequilibrium with the main components
			<ref>Ernst Carl Gerlach Stueckelberg "Theoreme H et unitarite de S", Helvetica Physica Acta '''25''' pp. 577-580 (1952)</ref>.
			Under these microscopic assumptions, the semi-detailed balance condition becomes the balance equation for the Markov microkinetics according to the Michaelis-Menten-Stueckelberg theorem<ref name="GorbanShahzad2011">[http://dx.doi.org/10.3390/e13050966 Alexander N. Gorban and Muhammad Shahzad "The Michaelis-Menten-Stueckelberg Theorem", Entropy '''13''' pp. 966-1019 (2011)]</ref>.

	== Dissipation in systems with semi-detailed balance ==		== Dissipation in systems with semi-detailed balance ==

Carl McBride: Slight tidy

2012-02-08T14:54:43Z

Slight tidy

Carl McBride: /* Dissipation in systems with detailed balance */ Further tidy

2012-02-08T14:38:11Z

Dissipation in systems with detailed balance: Further tidy

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	==Dissipation in systems with detailed balance==		==Dissipation in systems with detailed balance==
	To describe dynamics of the systems that obey the generalised mass action law, one has to represent the activities as functions of the concentrations <math>c_j</math> and [[temperature]]. For this purpose, let us represent the activity in terms of the [[chemical potential]]:		To describe dynamics of the systems that obey the generalised mass action law, one has to represent the activities as functions of the concentrations <math>c_j</math> and the [[temperature]]. For this purpose, let us represent the activity in terms of the [[chemical potential]]:

	:<math>a_i = \exp\left (\frac{\mu_i - \mu^{\~~ominus~~}_i}{RT}\right )</math>		:<math>a_i = \exp\left (\frac{\mu_i - \mu^{\Theta}_i}{RT}\right )</math>

	where <math>\mu_i</math> is the chemical potential of the species under the conditions of interest, ~~''μ''~~<~~sup~~>~~<s>o~~</~~s></sup><sub>''i''</sub~~> is the chemical potential of that species in the chosen standard state, ''R'' is the gas constant and ''T'' is the thermodynamic temperature.		where <math>\mu_i</math> is the chemical potential of the species under the conditions of interest, <math>\mu^{\Theta}_i</math> is the chemical potential of that species in the chosen standard state, <math>R</math> is the [[Molar gas constant \|gas constant]] and <math>T</math> is the thermodynamic temperature.
	The chemical potential can be represented as a function of ''c'' and ''T'', where ''c'' is the vector of concentrations with components ~~''c~~<~~sub~~>j</~~sub~~>''. For ~~the~~ ideal systems, <math>\mu_i=RT\ln c_i+\mu^{\~~ominus~~}_i</math> and <math>a_j=c_j</math>: the activity is the concentration ~~and the generalized mass action law is the usual law of mass action~~.		The chemical potential can be represented as a function of <math>c</math> and <math>T</math>, where <math>c</math> is the vector of concentrations with components <math>c_j</math>. For ideal systems, <math>\mu_i=RT\ln c_i+\mu^{\Theta}_i</math> and <math>a_j=c_j</math>: the activity is the concentration.

	~~Let us consider a system in~~ isothermal ~~(''T''=const)~~ isochoric ~~(the volume ''V''=const) condition. For these conditions,~~ the Helmholtz ~~free~~ energy ~~''F~~(T,V,N)'' measures the ~~“useful”~~ work obtainable from a system. It is a functions of the temperature ''T'', the volume ''V'' and the amounts of chemical components ''N<sub>j</sub>'' (usually measured in moles), ''N'' is the vector with components ''N<sub>j</sub>''. For ~~the~~ ideal ~~systems, <math>F=RT \sum_i N_i \left(\ln\left(\frac{N_i}{V}\right)-1+\frac{\mu^{\ominus}_i(T)}{RT}\right) </math>~~		For an [[Canonical ensemble \|isothermal, isochoric system]] the [[Helmholtz energy function]] <math>A(N, V, T)</math> measures the "useful" [[work]] obtainable from a system. For an ideal system one has

	The chemical potential is a partial derivative: <math> \mu_i=\partial F(T,V,N)/\partial N_j</math>.		:<math>A=RT \sum_i N_i \left(\ln\left(\frac{N_i}{V}\right)-1+\frac{\mu^{\Theta}_i(T)}{RT}\right) </math>

			The chemical potential is given by the partial derivative

			:<math> \mu_i=\frac{\partial A(N,V,T)}{\partial N_j}.</math>

	The chemical kinetic equations are		The chemical kinetic equations are

	:<math>\frac{d N_i}{d t}=V \sum_r \gamma_{ri}(w^+_r-w^-_r) .</math>		:<math>\frac{d N_i}{d t}=V \sum_r \gamma_{ri}(w^+_r-w^-_r) .</math>

	If the principle of detailed balance is valid then for any value of ''T'' there exists a positive point of detailed balance ~~''c''~~<~~sup~~>eq</~~sup~~>:		If the principle of detailed balance is valid, then for any value of <math>T</math> there exists a positive point of detailed balance <math>c^{\mathrm {eq}}</math>:
	:<math>w^+_r(c^{\rm eq},T)=w^-_r(c^{\rm eq},T)=w^{\rm eq}_r</math>
	~~Elementary algebra gives~~		:<math>w^+_r(c^{\mathrm {eq}},T)=w^-_r(c^{\rm eq},T)=w^{\rm eq}_r</math>

			which leads to

	:<math>w^+_r=w^{\rm eq}_r \exp \left(\sum_i \frac{\alpha_{ri}(\mu_i-\mu^{\rm eq}_i)}{RT}\right); \;\; w^-_r=w^{\rm eq}_r \exp \left(\sum_i \frac{\beta_{ri}(\mu_i-\mu^{\rm eq}_i)}{RT}\right);</math>		:<math>w^+_r=w^{\rm eq}_r \exp \left(\sum_i \frac{\alpha_{ri}(\mu_i-\mu^{\rm eq}_i)}{RT}\right); \;\; w^-_r=w^{\rm eq}_r \exp \left(\sum_i \frac{\beta_{ri}(\mu_i-\mu^{\rm eq}_i)}{RT}\right);</math>

	where <math>\mu^{\rm eq}_i=\mu_i(c^{\rm eq},T)</math>		where <math>\mu^{\rm eq}_i=\mu_i(c^{\rm eq},T)</math>

	For the dissipation ~~we obtain from these formulas:~~		For the dissipation one obtains

	:<math>\frac{d F}{d t}=\sum_i \frac{\partial F(T,V,N)}{\partial N_i} \frac{d N_i}{d t}=\sum_i \mu_i \frac{d N_i}{d t} = -VRT \sum_r (\ln w_r^+-\ln w_r^-) (w_r^+-w_r^-) \leq 0</math>		:<math>\frac{d F}{d t}=\sum_i \frac{\partial F(T,V,N)}{\partial N_i} \frac{d N_i}{d t}=\sum_i \mu_i \frac{d N_i}{d t} = -VRT \sum_r (\ln w_r^+-\ln w_r^-) (w_r^+-w_r^-) \leq 0</math>
	~~The inequality holds because ln is a monotone function and, hence, the expressions <math>\ln w_r^+-\ln w_r^-</math> and <math>w_r^+-w_r^-</math> have always 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 conditions the Gibbs ~~free~~ energy decreases, for ~~the~~ isochoric systems with ~~the~~ constant internal energy ~~(isolated systems)~~ the entropy increases as well as for isobaric systems with the constant enthalpy.		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 isobaric systems with the constant [[enthalpy]].

	== Onsager reciprocal relations and detailed balance ==		== Onsager reciprocal relations and detailed balance ==

Carl McBride: Further tidy

2012-02-08T14:13:24Z

Further tidy

← Older revision		Revision as of 16:13, 8 February 2012
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	In 1901, R. Wegscheider introduced the principle of detailed balance for chemical kinetics.<ref>[http://dx.doi.org/10.1007/BF01517498 Rud Wegscheider "Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme", Monatshefte für Chemie '''22''' pp. 849-906 (1901)]</ref> In particular, he demonstrated that the irreversible cycles <math>A_1 \to A_2 \to ... \to A_n \to A_1</math> are impossible and found explicitly the relations between kinetic constants that follow from the principle of detailed balance.		In 1901, R. Wegscheider introduced the principle of detailed balance for chemical kinetics.<ref>[http://dx.doi.org/10.1007/BF01517498 Rud Wegscheider "Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme", Monatshefte für Chemie '''22''' pp. 849-906 (1901)]</ref> In particular, he demonstrated that the irreversible cycles <math>A_1 \to A_2 \to ... \to A_n \to A_1</math> are impossible and found explicitly the relations between kinetic constants that follow from the principle of detailed balance.

	See also <ref name=vanKampen1992>van Kampen, N.G. "Stochastic Processes in Physics and Chemistry", Elsevier Science (1992).</ref>~~<ref name=Yab1991>Yablonskii, G.S., Bykov, V.I., Gorban, A.N., Elokhin, V.I. (1991), Kinetic Models of Catalytic Reactions, Amsterdam, The Netherlands: Elsevier.</ref>~~<ref>Lifshitz, E.M., Pitaevskii, L.P., Physical kinetics (1981)London: Pergamon. Vol. 10 of the Course of Theoretical Physics(3rd Ed).</ref>		See also <ref name=vanKampen1992>van Kampen, N.G. "Stochastic Processes in Physics and Chemistry", Elsevier Science (1992).</ref>
			<ref>Lifshitz, E.M., Pitaevskii, L.P., Physical kinetics (1981)London: Pergamon. Vol. 10 of the Course of Theoretical Physics(3rd Ed).</ref>

	==Microscopic background==		==Microscopic background==
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	The ''stoichiometric matrix'' is <math>\boldsymbol{\Gamma}=(\gamma_{ri})</math>, <math>\gamma_{ri}=\beta_{ri}-\alpha_{ri}</math> (gain minus loss). The ''stoichiometric vector'' <math>\gamma_r</math> is the ''r''th row of <math>\boldsymbol{\Gamma}</math> with coordinates <math>\gamma_{ri}=\beta_{ri}-\alpha_{ri}</math>.		The ''stoichiometric matrix'' is <math>\boldsymbol{\Gamma}=(\gamma_{ri})</math>, <math>\gamma_{ri}=\beta_{ri}-\alpha_{ri}</math> (gain minus loss). The ''stoichiometric vector'' <math>\gamma_r</math> is the ''r''th row of <math>\boldsymbol{\Gamma}</math> with coordinates <math>\gamma_{ri}=\beta_{ri}-\alpha_{ri}</math>.

	According to the ''~~generalized~~ mass action law'', the reaction rate for an elementary reaction is		According to the ''generalised mass action law'', the reaction rate for an elementary reaction is

	:<math>w_r=k_r \prod_{i=1}^n a_i^{\alpha_{ri}} \, ,</math>		:<math>w_r=k_r \prod_{i=1}^n a_i^{\alpha_{ri}} \, ,</math>
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	<math>\sum_i \gamma_{ri} x_i = \ln k_r^+-\ln k_r^-=\ln K_r </math>		<math>\sum_i \gamma_{ri} x_i = \ln k_r^+-\ln k_r^-=\ln K_r </math>

	is solvable (<math>x_i=\ln a_i^{\rm eq}</math>). The following classical result gives the necessary and sufficient conditions for the existence of the positive equilibrium <math>a_i^{\rm eq}>0</math> with detailed balance ~~(see, for example, the textbook~~<ref name=Yab1991/>).		is solvable (<math>x_i=\ln a_i^{\rm eq}</math>). The following classical result gives the necessary and sufficient conditions for the existence of the positive equilibrium <math>a_i^{\rm eq}>0</math> with detailed balance <ref name=Yab1991>G. S. Yablonskii, V. I. Bykov, V. I. Elokhin, and A.N. Gorban "Kinetic Models of Catalytic Reactions", Elsevier (1991) ISBN 978-0-444-88802-0</ref>.

	Two conditions are sufficient and necessary ~~for solvability of~~ the system of detailed balance equations:		Two conditions are sufficient and necessary in order to solve the system of detailed balance equations:
	# If <math>k_r^+>0</math> then <math>k_r^->0</math> (reversibility);		# If <math>k_r^+>0</math> then <math>k_r^->0</math> (reversibility);
	# For any solution <math>\boldsymbol{\lambda}=(\lambda_r)</math> of the system		# For any solution <math>\boldsymbol{\lambda}=(\lambda_r)</math> of the system
	:<math>\boldsymbol{\lambda \Gamma} =0 \;\; \left(\mbox{i.e.}\;\; \sum_r \lambda_r \gamma_{ri}=0\;\; \mbox{for all} \;\; i\right)</math>		:<math>\boldsymbol{\lambda \Gamma} =0 \;\; \left(\mbox{i.e.}\;\; \sum_r \lambda_r \gamma_{ri}=0\;\; \mbox{for all} \;\; i\right)</math>
	the Wegscheider's identity<ref name=GorbanYablonsky2011>~~Gorban, A.N., Yablonsky, G.S. (2011)~~ [http://~~arxiv~~.org/~~PS_cache~~/~~arxiv/pdf/1101/1101~~.~~5280v3~~.~~pdf~~ Extended detailed balance for systems with irreversible reactions], Chemical Engineering Science 66~~, 5388–5399~~.</ref> holds:		the Wegscheider's identity
			<ref name=GorbanYablonsky2011>[http://dx.doi.org/10.1016/j.ces.2011.07.054 A. N. Gorban and G. S. Yablonsky "Extended detailed balance for systems with irreversible reactions", Chemical Engineering Science '''66''' pp. 5388-5399 (2011)]</ref>
			holds:
	:<math>\prod_{r=1}^m (k_r^+)^{\lambda_r}=\prod_{r=1}^m (k_r^-)^{\lambda_r} \, .</math>		:<math>\prod_{r=1}^m (k_r^+)^{\lambda_r}=\prod_{r=1}^m (k_r^-)^{\lambda_r} \, .</math>

	''Remark.'' It is sufficient to use in the Wegscheider conditions a basis of solutions of the system <math>\boldsymbol{\lambda \Gamma} =0 </math>.		''Remark.'' It is sufficient to use in the Wegscheider conditions a basis of solutions of the system <math>\boldsymbol{\lambda \Gamma} =0 </math>.

	In particular, for any cycle in the monomolecular (linear) reactions the product of the reaction rate constants in the clockwise direction is equal to the product of the reaction rate constants in the ~~counterclockwise~~ direction. The same condition is valid for the reversible Markov processes (it is equivalent to the "no net flow" condition).		In particular, for any cycle in the monomolecular (linear) reactions the product of the reaction rate constants in the clockwise direction is equal to the product of the reaction rate constants in the counter-clockwise direction. The same condition is valid for the reversible Markov processes (it is equivalent to the "no net flow" condition).

	A simple ~~nonlinear~~ example gives us a linear cycle supplemented by one ~~nonlinear~~ step<ref name=GorbanYablonsky2011/>:		A simple non-linear example gives us a linear cycle supplemented by one non-linear step <ref name=GorbanYablonsky2011/>:
	# <math>A_1 \rightleftharpoons A_2</math>		# <math>A_1 \rightleftharpoons A_2</math>
	# <math>A_2 \rightleftharpoons A_3 </math>		# <math>A_2 \rightleftharpoons A_3 </math>
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	:<math>\gamma_1+\gamma_2+\gamma_3=0</math> and <math>\gamma_3+\gamma_4-\gamma_2=0</math>.		:<math>\gamma_1+\gamma_2+\gamma_3=0</math> and <math>\gamma_3+\gamma_4-\gamma_2=0</math>.

	The computational aspect of the Wegscheider conditions was studied by D. Colquhoun ~~with co-authors~~.<ref>Colquhoun, D., Dowsland, ~~K.A.,~~ Beato~~, M.,~~ and ~~Plested, A.~~J.R. ~~(2004) [http://www.ucl.ac.uk/Pharmacology/dc-bits/colquhoun-biophysj-04.pdf~~ How to Impose Microscopic Reversibility in Complex Reaction Mechanisms], Biophysical 86~~, June~~ 2004~~, 3510–3518~~</ref>		The computational aspect of the Wegscheider conditions was studied by D. Colquhoun et al.
			<ref>[http://dx.doi.org/10.1529/biophysj.103.038679 David Colquhoun, Kathryn A. Dowsland, Marco Beato and Andrew J.R. Plested "How to Impose Microscopic Reversibility in Complex Reaction Mechanisms", Biophysical Journal '''86''' pp. 3510-3518 (2004)]</ref>.

	The Wegscheider conditions demonstrate that whereas the principle of detailed balance states a local property of equilibrium, it implies the relations between the kinetic constants that are valid for all states far from equilibrium. This is possible because a kinetic law is known and relations between the rates of the elementary processes at equilibrium can be transformed into relations between kinetic constants which are used globally. For the Wegscheider conditions this kinetic law is the law of mass action (or the ~~generalized~~ law of mass action).		The Wegscheider conditions demonstrate that, whereas the principle of detailed balance states a local property of equilibrium, it implies the relations between the kinetic constants that are valid for all states far from equilibrium. This is possible because a kinetic law is known and relations between the rates of the elementary processes at equilibrium can be transformed into relations between kinetic constants which are used globally. For the Wegscheider conditions this kinetic law is the law of mass action (or the generalised law of mass action).

	==Dissipation in systems with detailed balance==		==Dissipation in systems with detailed balance==
			To describe dynamics of the systems that obey the generalised mass action law, one has to represent the activities as functions of the concentrations <math>c_j</math> and [[temperature]]. For this purpose, let us represent the activity in terms of the [[chemical potential]]:

	To describe dynamics of the systems that obey the generalized mass action law, one has to represent the activities as functions of the concentrations ''c<sub>j</sub>'' and temperature. For this purpose, let us the representation of the activity through the chemical potential:
	:<math>a_i = \exp\left (\frac{\mu_i - \mu^{\ominus}_i}{RT}\right )</math>		:<math>a_i = \exp\left (\frac{\mu_i - \mu^{\ominus}_i}{RT}\right )</math>
	where ~~''μ~~<~~sub~~>i</~~sub~~>'' is the chemical potential of the species under the conditions of interest, ''μ''<sup><s>o</s></sup><sub>''i''</sub> is the chemical potential of that species in the chosen standard state, ''R'' is the gas constant and ''T'' is the thermodynamic temperature.
			where <math>\mu_i</math> is the chemical potential of the species under the conditions of interest, ''μ''<sup><s>o</s></sup><sub>''i''</sub> is the chemical potential of that species in the chosen standard state, ''R'' is the gas constant and ''T'' is the thermodynamic temperature.
	The chemical potential can be represented as a function of ''c'' and ''T'', where ''c'' is the vector of concentrations with components ''c<sub>j</sub>''. For the ideal systems, <math>\mu_i=RT\ln c_i+\mu^{\ominus}_i</math> and <math>a_j=c_j</math>: the activity is the concentration and the generalized mass action law is the usual law of mass action.		The chemical potential can be represented as a function of ''c'' and ''T'', where ''c'' is the vector of concentrations with components ''c<sub>j</sub>''. For the ideal systems, <math>\mu_i=RT\ln c_i+\mu^{\ominus}_i</math> and <math>a_j=c_j</math>: the activity is the concentration and the generalized mass action law is the usual law of mass action.

Carl McBride: /* Detailed balance and the entropy growth */ Slight tidy of references

2012-02-08T13:53:32Z

Detailed balance and the entropy growth: Slight tidy of references

← Older revision		Revision as of 15:53, 8 February 2012
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	==Detailed balance and the entropy growth==		==Detailed balance and the entropy growth==
	For many systems of physical and chemical kinetics, detailed balance provides ''sufficient conditions'' for the [[entropy]] growth in isolated systems. For example, the famous Boltzmann H-theorem<ref name = "Boltzmann1872"/> states that, according to the Boltzmann equation, the principle of detailed balance implies positivity of the entropy production. The Boltzmann formula (1872) for the entropy production in the rarefied gas kinetics with detailed balance<ref name = "Boltzmann1872"/><ref name=Tolman1938/> served as a prototype of many similar formulas for dissipation in mass action kinetics<ref>~~Volpert,~~ A.I.~~, Khudyaev,~~ S.I. ~~(1985),~~ Analysis in classes of discontinuous functions and equations of mathematical physics~~. Dordrecht~~, ~~The Netherlands: Nijoff.~~ (~~Translation from the 1st Russian ed., Moscow, Nauka publ., 1975.~~)</ref> and ~~generalized~~ mass action kinetics<ref>~~Schuster, S., Schuster R. (1989).~~ [http://~~www~~.~~springerlink~~.~~com~~/~~content/g3m2177v4344065q~~/ A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation~~.] J. Math. Chem~~, 3 ~~(1),~~ 25-42.</ref> with detailed balance.		For many systems that treat physical and chemical kinetics, detailed balance provides ''sufficient conditions'' for the [[entropy]] growth in isolated systems. For example, the famous Boltzmann H-theorem<ref name = "Boltzmann1872"/> states that, according to the [[Boltzmann equation]], the principle of detailed balance implies positivity of the entropy production. The Boltzmann formula (1872) for the entropy production in the rarefied gas kinetics with detailed balance<ref name = "Boltzmann1872"/><ref name=Tolman1938/> served as a prototype of many similar formulas for dissipation in mass action kinetics
			<ref>A.I. Volpert and S. I. Khudyaev "Analysis in classes of discontinuous functions and equations of mathematical physics", Springer (1985) ISBN 978-90-247-3109-1 </ref>
			and generalised mass action kinetics
			<ref>[http://dx.doi.org/10.1007/BF01171883 Stefan Schuster and Ronny Schuster "A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation", Journal of Mathematical Chemistry '''3''' pp. 25-42 (1989)]</ref>
			with detailed balance.

	Nevertheless, the principle of detailed balance is not necessary for ~~the~~ entropy growth. For example, in the linear irreversible cycle <math>A_1 \to A_2 \to A_3 \to A_1</math> the entropy production is positive but the principle of detailed balance does not hold.		Nevertheless, the principle of detailed balance is not necessary for entropy growth. For example, in the linear irreversible cycle <math>A_1 \to A_2 \to A_3 \to A_1</math> the entropy production is positive but the principle of detailed balance does not hold.

	The principle of detailed balance is a ''sufficient but not necessary condition'' for the entropy growth in the Boltzmann kinetics. These relations between the principle of detailed balance and the Second law of thermodynamics were clarified in 1887 when Hendrik Lorentz objected the Boltzmann H-theorem for polyatomic gases.<ref>Lorentz ~~H.-A. (1887)~~ Üeber das Gleichgewicht der lebendigen Kraft unter Gasmolekülen. S.A.W. [Sitzungsberichte der Kgl. Akademie der Wissenschaften in Wien. Vienne.~~] 95,~~ 115-152.</ref> Lorentz stated that the principle of detailed balance is not applicable to collisions of polyatomic molecules. Boltzmann immediately invented a new, more general condition sufficient for the entropy growth.<ref name=Boltzmann1887>~~Boltzmann~~ L. ~~(1887)~~ Neuer Beweis zweier Sätze über das Wärmegleichgewicht unter mehratomigen Gasmolekülen. S.A.W. [Sitzungsberichte der Kgl. Akademie der Wissenschaften in Wien. Vienne.~~] 95,~~ 153-164.</ref> In particular, this condition is valid for all Markov processes without any relation to time-reversibility. The entropy growth in all Markov processes was ~~explicitely~~ proved later.<ref>~~Shannon, C.E. (1948) A Mathematical Theory of Communication, Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656.~~ [http://www.alcatel-lucent.com/bstj/vol27-1948/articles/bstj27-3-379.pdf] [http://www.alcatel-lucent.com/bstj/vol27-1948/articles/bstj27-4-623.pdf]</ref> These theorems may be considered as simplifications of the Boltzmann result. Later, this condition was discussed as the "cyclic balance" condition (because it holds for irreversible cycles) or the "semi-detailed balance" or the "complex balance". In 1981, Carlo Cercignani and Maria Lampis proved that the Lorenz arguments were wrong and the principle of detailed balance is valid for polyatomic molecules.<ref>Cercignani~~, C.~~ and Lampis~~, M. (1981).~~ On the H-theorem for polyatomic gases, Journal of Statistical Physics~~, V~~. ~~26 (4),~~ 795-801.</ref> Nevertheless, the extended semi-detailed balance conditions invented by Boltzmann in this discussion remain the ~~remarkable generalization~~ of ~~the~~ detailed balance.		The principle of detailed balance is a ''sufficient but not necessary condition'' for the entropy growth in the Boltzmann kinetics. These relations between the principle of detailed balance and the [[Second law of thermodynamics]] were clarified in 1887 when Hendrik Lorentz objected the Boltzmann H-theorem for polyatomic gases
			<ref>Hendrik Antoon Lorentz "Üeber das Gleichgewicht der lebendigen Kraft unter Gasmolekülen" S.A.W. (Sitzungsberichte der Kgl. Akademie der Wissenschaften in Wien. Vienne) '''95''' pp. 115-152 (1887)</ref>.
			Lorentz stated that the principle of detailed balance is not applicable to collisions of polyatomic molecules. Boltzmann immediately invented a new, more general condition sufficient for the entropy growth.
			<ref name=Boltzmann1887>L. Boltzmann "Neuer Beweis zweier Sätze über das Wärmegleichgewicht unter mehratomigen Gasmolekülen" S.A.W. (Sitzungsberichte der Kgl. Akademie der Wissenschaften in Wien. Vienne) '''95''' pp. 153-164 (1887)</ref>.
			In particular, this condition is valid for all Markov processes without any relation to time-reversibility. The entropy growth in all Markov processes was explicitly proved later
			<ref>[http://www.alcatel-lucent.com/bstj/vol27-1948/articles/bstj27-3-379.pdf Claude Elwood Shannon "A Mathematical Theory of Communication", Bell System Technical Journal '''27''' July pp. 379-423 (1948)]</ref>
			<ref>[http://www.alcatel-lucent.com/bstj/vol27-1948/articles/bstj27-4-623.pdf Claude Elwood Shannon "A Mathematical Theory of Communication", Bell System Technical Journal '''27''' October pp. 623-656 (1948)]</ref>.
			These theorems may be considered as simplifications of the Boltzmann result. Later, this condition was discussed as the "cyclic balance" condition (because it holds for irreversible cycles) or the "semi-detailed balance" or the "complex balance". In 1981, Carlo Cercignani and Maria Lampis proved that the Lorenz arguments were wrong and the principle of detailed balance is valid for polyatomic molecules
			<ref>[http://dx.doi.org/10.1007/BF01010940 Carlo Cercignani and Maria Lampis "On the H-theorem for polyatomic gases", Journal of Statistical Physics '''26''' pp. 795-801 (1981)]</ref>.
			Nevertheless, the extended semi-detailed balance conditions invented by Boltzmann in this discussion remain the generalisation of detailed balance.

	==Wegscheider's conditions for the generalized mass action law==		==Wegscheider's conditions for the generalized mass action law==

Carl McBride: Placed region of text in a blockquote

2012-02-07T16:48:13Z

Placed region of text in a blockquote

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	<blockquote>''Corresponding to every individual process there is a reverse process, and in a state of equilibrium the average rate of every process is equal to the average rate of its reverse process.<ref>[http://www.pnas.org/content/11/3/179.full.pdf+html Gilbert N. Lewis "A new principle of equilibrium" Proceedings of the National Academy of Sciences '''11''' pp. 179-183 (1925)]''</ref></blockquote>		<blockquote>''Corresponding to every individual process there is a reverse process, and in a state of equilibrium the average rate of every process is equal to the average rate of its reverse process.<ref>[http://www.pnas.org/content/11/3/179.full.pdf+html Gilbert N. Lewis "A new principle of equilibrium" Proceedings of the National Academy of Sciences '''11''' pp. 179-183 (1925)]''</ref></blockquote>

	According to Ter Haar <ref>[http://dx.doi.org/10.1103/RevModPhys.27.289 D. Ter Haar "Foundations of Statistical Mechanics", Reviews of Modern Physics '''27''' pp. 289-338 (1955)] Appendix VII</ref> the essence of the detailed balance is: "...at equilibrium the number of processes destroying situation <math>A</math> and creating situation <math>B</math> will be equal to the number of processes producing <math>A</math> and destroying <math>B</math>.		According to Ter Haar <ref>[http://dx.doi.org/10.1103/RevModPhys.27.289 D. Ter Haar "Foundations of Statistical Mechanics", Reviews of Modern Physics '''27''' pp. 289-338 (1955)] Appendix VII</ref> the essence of the detailed balance is: <blockquote>...at equilibrium the number of processes destroying situation <math>A</math> and creating situation <math>B</math> will be equal to the number of processes producing <math>A</math> and destroying <math>B</math></blockquote>

	The principle of detailed balance was explicitly introduced for collisions by [[Ludwig Eduard Boltzmann \|Ludwig Boltzmann]]. In 1872, he proved his [[H-theorem]] using this principle <ref name = "Boltzmann1872">Ludwig Boltzmann "Lectures on Gas Theory" (original title "Vorlesungen uber Gastheorie") (1896) ISBN 0486684555 </ref>.		The principle of detailed balance was explicitly introduced for collisions by [[Ludwig Eduard Boltzmann \|Ludwig Boltzmann]]. In 1872, he proved his [[H-theorem]] using this principle <ref name = "Boltzmann1872">Ludwig Boltzmann "Lectures on Gas Theory" (original title "Vorlesungen uber Gastheorie") (1896) ISBN 0486684555 </ref>.

Carl McBride: /* Reversible Markov chains */ I think the publisher is Wiley not Oxford

2012-02-07T16:45:08Z

Reversible Markov chains: I think the publisher is Wiley not Oxford

← Older revision		Revision as of 18:45, 7 February 2012
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	:<math>\pi_{i} P_{ij} = \pi_{j} P_{ji}\,,</math>		:<math>\pi_{i} P_{ij} = \pi_{j} P_{ji}\,,</math>

	where <math>P</math> is the Markov transition matrix (transition probability), ''i.e.'', ''P''<sub>''ij''</sub> = ''P''(''X''<sub>''t''</sub> = ''j'' \| ''X''<sub>''t'' − 1</sub> = ''i''); and π<sub>''i''</sub> and π<sub>''j''</sub> are the equilibrium probabilities of being in states ''i'' and ''j'', respectively.<ref name=OHagan>Anthony O'Hagan and Jonathan J. Forster "Kendall's Advanced Theory of Statistics, Volume 2B: Bayesian Inference", ~~Oxford University Press~~ (2004) ISBN 0340807520</ref> When Pr(''X''<sub>''t''−1</sub> = ''i'') = π<sub>''i''</sub> for all ''i'', this is equivalent to the joint probability matrix, Pr(''X''<sub>''t''−1</sub> = ''i'', ''X''<sub>''t''</sub> = ''j'') being symmetric in ''i'' and ''j''; or symmetric in ''t'' − 1 and ''t''.		where <math>P</math> is the Markov transition matrix (transition probability), ''i.e.'', ''P''<sub>''ij''</sub> = ''P''(''X''<sub>''t''</sub> = ''j'' \| ''X''<sub>''t'' − 1</sub> = ''i''); and π<sub>''i''</sub> and π<sub>''j''</sub> are the equilibrium probabilities of being in states ''i'' and ''j'', respectively.<ref name=OHagan>Anthony O'Hagan and Jonathan J. Forster "Kendall's Advanced Theory of Statistics, Volume 2B: Bayesian Inference", Wiley (2004) ISBN 0340807520</ref> When Pr(''X''<sub>''t''−1</sub> = ''i'') = π<sub>''i''</sub> for all ''i'', this is equivalent to the joint probability matrix, Pr(''X''<sub>''t''−1</sub> = ''i'', ''X''<sub>''t''</sub> = ''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 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'':

Carl McBride: /* Reversible Markov chains */ Changed format of cite/reference

2012-02-07T16:43:46Z

Reversible Markov chains: Changed format of cite/reference

← Older revision		Revision as of 18:43, 7 February 2012
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	:<math>\pi_{i} P_{ij} = \pi_{j} P_{ji}\,,</math>		:<math>\pi_{i} P_{ij} = \pi_{j} P_{ji}\,,</math>

	where <math>P</math> is the Markov transition matrix (transition probability), ''i.e.'', ''P''<sub>''ij''</sub> = ''P''(''X''<sub>''t''</sub> = ''j'' \| ''X''<sub>''t'' − 1</sub> = ''i''); and π<sub>''i''</sub> and π<sub>''j''</sub> are the equilibrium probabilities of being in states ''i'' and ''j'', respectively.<ref name=OHagan>O'Hagan~~, A~~., Forster~~, J. (2004)~~ Kendall's Advanced Theory of Statistics, Volume 2B: Bayesian Inference. Oxford University Press~~, New York.~~</ref> When Pr(''X''<sub>''t''−1</sub> = ''i'') = π<sub>''i''</sub> for all ''i'', this is equivalent to the joint probability matrix, Pr(''X''<sub>''t''−1</sub> = ''i'', ''X''<sub>''t''</sub> = ''j'') being symmetric in ''i'' and ''j''; or symmetric in ''t'' − 1 and ''t''.		where <math>P</math> is the Markov transition matrix (transition probability), ''i.e.'', ''P''<sub>''ij''</sub> = ''P''(''X''<sub>''t''</sub> = ''j'' \| ''X''<sub>''t'' − 1</sub> = ''i''); and π<sub>''i''</sub> and π<sub>''j''</sub> are the equilibrium probabilities of being in states ''i'' and ''j'', respectively.<ref name=OHagan>Anthony O'Hagan and Jonathan J. Forster "Kendall's Advanced Theory of Statistics, Volume 2B: Bayesian Inference", Oxford University Press (2004) ISBN 0340807520</ref> When Pr(''X''<sub>''t''−1</sub> = ''i'') = π<sub>''i''</sub> for all ''i'', this is equivalent to the joint probability matrix, Pr(''X''<sub>''t''−1</sub> = ''i'', ''X''<sub>''t''</sub> = ''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 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'':

Carl McBride: Further tidy

2012-02-07T16:31:05Z

Further tidy

← Older revision		Revision as of 18:31, 7 February 2012
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	The principle of detailed balance was explicitly introduced for collisions by [[Ludwig Eduard Boltzmann \|Ludwig Boltzmann]]. In 1872, he proved his [[H-theorem]] using this principle <ref name = "Boltzmann1872">Ludwig Boltzmann "Lectures on Gas Theory" (original title "Vorlesungen uber Gastheorie") (1896) ISBN 0486684555 </ref>.		The principle of detailed balance was explicitly introduced for collisions by [[Ludwig Eduard Boltzmann \|Ludwig Boltzmann]]. In 1872, he proved his [[H-theorem]] using this principle <ref name = "Boltzmann1872">Ludwig Boltzmann "Lectures on Gas Theory" (original title "Vorlesungen uber Gastheorie") (1896) ISBN 0486684555 </ref>.
	The arguments in favour of this property are founded upon microscopic reversibility <ref name=Tolman1938>Richard C. Tolman "The Principles of Statistical Mechanics" Oxford University Press (1938) ISBN 0486638960</ref>.		The arguments in favour of this property are founded upon microscopic reversibility <ref name=Tolman1938>Richard C. Tolman "The Principles of Statistical Mechanics" Oxford University Press (1938) ISBN 0486638960</ref>.
	In 1901, R. Wegscheider introduced the principle of detailed balance for chemical kinetics.<ref>[http://dx.doi.org/10.1007/BF01517498 Rud Wegscheider "Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme", Monatshefte für Chemie '''22''' pp. 849-906 (1901)]</ref> In particular, he demonstrated that the irreversible cycles <math>A_1 \to A_2 \to ... \to A_n \to A_1</math> are impossible and found explicitly the relations between kinetic constants that follow from the principle of detailed balance. In 1931, Lars Onsager used these relations in his works<ref name = "Onsager1931">Onsager, L. (1931), [http://prola.aps.org/abstract/PR/v37/i4/p405_1 Reciprocal relations in irreversible processes.] I, Phys. Rev. 37, 405-426; II 38, 2265-2279</ref>, for which he was awarded the 1968 Nobel Prize in Chemistry.		In 1901, R. Wegscheider introduced the principle of detailed balance for chemical kinetics.<ref>[http://dx.doi.org/10.1007/BF01517498 Rud Wegscheider "Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme", Monatshefte für Chemie '''22''' pp. 849-906 (1901)]</ref> In particular, he demonstrated that the irreversible cycles <math>A_1 \to A_2 \to ... \to A_n \to A_1</math> are impossible and found explicitly the relations between kinetic constants that follow from the principle of detailed balance.

	See also <ref name=vanKampen1992>van Kampen, N.G. "Stochastic Processes in Physics and Chemistry", Elsevier Science (1992).</ref><ref name=Yab1991>Yablonskii, G.S., Bykov, V.I., Gorban, A.N., Elokhin, V.I. (1991), Kinetic Models of Catalytic Reactions, Amsterdam, The Netherlands: Elsevier.</ref><ref>Lifshitz, E.M., Pitaevskii, L.P., Physical kinetics (1981)London: Pergamon. Vol. 10 of the Course of Theoretical Physics(3rd Ed).</ref>		See also <ref name=vanKampen1992>van Kampen, N.G. "Stochastic Processes in Physics and Chemistry", Elsevier Science (1992).</ref><ref name=Yab1991>Yablonskii, G.S., Bykov, V.I., Gorban, A.N., Elokhin, V.I. (1991), Kinetic Models of Catalytic Reactions, Amsterdam, The Netherlands: Elsevier.</ref><ref>Lifshitz, E.M., Pitaevskii, L.P., Physical kinetics (1981)London: Pergamon. Vol. 10 of the Course of Theoretical Physics(3rd Ed).</ref>

	==Microscopic background==		==Microscopic background==
			The microscopic "reversing of time" turns at the kinetic level into the "reversing of arrows": the elementary processes transform into their reverse processes. For example, the reaction

	~~The microscopic "reversing of time" turns at the kinetic level into the "reversing of arrows"~~: ~~the elementary processes transform~~ into ~~their reverse processes. For example, the reaction~~		:<math>\sum_i \alpha_i A_i \to \sum_j \beta_j B_j</math>

			transforms into

	:~~<math>\sum_i \alpha_i A_i \to \sum_j \beta_j B_j</math> transforms into~~ <math>\sum_j \beta_j B_j \to \sum_i \alpha_i A_i</math>		:<math>\sum_j \beta_j B_j \to \sum_i \alpha_i A_i</math>

	and conversely. (Here, <math>A_i, B_j</math> are symbols of components or states, <math>\alpha_i, \beta_j \geq 0 </math> are coefficients). The equilibrium ensemble should be invariant with respect to this transformation because of ~~microreversibility~~ and the uniqueness of thermodynamic equilibrium. This leads us immediately to the concept of detailed balance: each process is equilibrated by its reverse process.		and conversely. (Here, <math>A_i, B_j</math> are symbols of components or states, <math>\alpha_i, \beta_j \geq 0 </math> are coefficients). The equilibrium ensemble should be invariant with respect to this transformation because of micro-reversibility and the uniqueness of thermodynamic equilibrium. This leads us immediately to the concept of detailed balance: each process is equilibrated by its reverse process.

	==Reversible Markov chains==		==Reversible Markov chains==
			A [[Markov chain \| Markov process]] is said to have detailed balance if the transition probability, <math>P</math>, between each pair of states <math>i</math> and <math>j</math> in the state space obey
	A Markov process is said to have detailed balance if the transition probability, ''P'', between each pair of states ''i'' and ''j'' in the state space obey

	:<math>\pi_{i} P_{ij} = \pi_{j} P_{ji}\,,</math>		:<math>\pi_{i} P_{ij} = \pi_{j} P_{ji}\,,</math>

	where ''P'' is the Markov transition matrix (transition probability), ''i.e.'', ''P''<sub>''ij''</sub> = ''P''(''X''<sub>''t''</sub> = ''j'' \| ''X''<sub>''t'' − 1</sub> = ''i''); and π<sub>''i''</sub> and π<sub>''j''</sub> are the equilibrium probabilities of being in states ''i'' and ''j'', respectively.<ref name=OHagan>O'Hagan, A., Forster, J. (2004) Kendall's Advanced Theory of Statistics, Volume 2B: Bayesian Inference. Oxford University Press, New York.</ref> When Pr(''X''<sub>''t''−1</sub> = ''i'') = π<sub>''i''</sub> for all ''i'', this is equivalent to the joint probability matrix, Pr(''X''<sub>''t''−1</sub> = ''i'', ''X''<sub>''t''</sub> = ''j'') being symmetric in ''i'' and ''j''; or symmetric in ''t'' − 1 and ''t''.		where <math>P</math> is the Markov transition matrix (transition probability), ''i.e.'', ''P''<sub>''ij''</sub> = ''P''(''X''<sub>''t''</sub> = ''j'' \| ''X''<sub>''t'' − 1</sub> = ''i''); and π<sub>''i''</sub> and π<sub>''j''</sub> are the equilibrium probabilities of being in states ''i'' and ''j'', respectively.<ref name=OHagan>O'Hagan, A., Forster, J. (2004) Kendall's Advanced Theory of Statistics, Volume 2B: Bayesian Inference. Oxford University Press, New York.</ref> When Pr(''X''<sub>''t''−1</sub> = ''i'') = π<sub>''i''</sub> for all ''i'', this is equivalent to the joint probability matrix, Pr(''X''<sub>''t''−1</sub> = ''i'', ''X''<sub>''t''</sub> = ''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 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'':
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	==Detailed balance and the entropy growth==		==Detailed balance and the entropy growth==
			For many systems of physical and chemical kinetics, detailed balance provides ''sufficient conditions'' for the [[entropy]] growth in isolated systems. For example, the famous Boltzmann H-theorem<ref name = "Boltzmann1872"/> states that, according to the Boltzmann equation, the principle of detailed balance implies positivity of the entropy production. The Boltzmann formula (1872) for the entropy production in the rarefied gas kinetics with detailed balance<ref name = "Boltzmann1872"/><ref name=Tolman1938/> served as a prototype of many similar formulas for dissipation in mass action kinetics<ref>Volpert, A.I., Khudyaev, S.I. (1985), Analysis in classes of discontinuous functions and equations of mathematical physics. Dordrecht, The Netherlands: Nijoff. (Translation from the 1st Russian ed., Moscow, Nauka publ., 1975.)</ref> and generalized mass action kinetics<ref>Schuster, S., Schuster R. (1989). [http://www.springerlink.com/content/g3m2177v4344065q/ A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation.] J. Math. Chem, 3 (1), 25-42.</ref> with detailed balance.
	For many systems of physical and chemical kinetics, detailed balance provides ''sufficient conditions'' for the entropy growth in isolated systems. For example, the famous Boltzmann H-theorem<ref name = "Boltzmann1872"/> states that, according to the Boltzmann equation, the principle of detailed balance implies positivity of the entropy production. The Boltzmann formula (1872) for the entropy production in the rarefied gas kinetics with detailed balance<ref name = "Boltzmann1872"/><ref name=Tolman1938/> served as a prototype of many similar formulas for dissipation in mass action kinetics<ref>Volpert, A.I., Khudyaev, S.I. (1985), Analysis in classes of discontinuous functions and equations of mathematical physics. Dordrecht, The Netherlands: Nijoff. (Translation from the 1st Russian ed., Moscow, Nauka publ., 1975.)</ref> and generalized mass action kinetics<ref>Schuster, S., Schuster R. (1989). [http://www.springerlink.com/content/g3m2177v4344065q/ A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation.] J. Math. Chem, 3 (1), 25-42.</ref> with detailed balance.

	Nevertheless, the principle of detailed balance is not necessary for the entropy growth. For example, in the linear irreversible cycle <math>A_1 \to A_2 \to A_3 \to A_1</math> the entropy production is positive but the principle of detailed balance does not hold.		Nevertheless, the principle of detailed balance is not necessary for the entropy growth. For example, in the linear irreversible cycle <math>A_1 \to A_2 \to A_3 \to A_1</math> the entropy production is positive but the principle of detailed balance does not hold.
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	:<math>\frac{d N_i}{d t}=-V \sum_j \left[\sum_r w^{\rm eq}_r \gamma_{ri}\gamma_{rj}\right] \frac{\mu_j-\mu^{\rm eq}_j}{RT}.</math>		:<math>\frac{d N_i}{d t}=-V \sum_j \left[\sum_r w^{\rm eq}_r \gamma_{ri}\gamma_{rj}\right] \frac{\mu_j-\mu^{\rm eq}_j}{RT}.</math>

	This is exactly the Onsager form: following the original work of Onsager,<ref ~~name =~~ "~~Onsager1931~~"/> we should introduce the thermodynamic forces <math>X_j</math> and the matrix of coefficients <math>L_{ij}</math> in the form		This is exactly the Onsager form: following the original work of [[Lars Onsager]] <ref>[http://dx.doi.org/10.1103/PhysRev.37.405 Lars Onsager "Reciprocal Relations in Irreversible Processes. I.", Physical Review '''37''' pp. 405-426 (1931)]</ref><ref>[http://dx.doi.org/10.1103/PhysRev.38.2265 Lars Onsager "Reciprocal Relations in Irreversible Processes. II.", Physical Review '''38''' pp. 2265-2279 (1931)]</ref>
			we should introduce the thermodynamic forces <math>X_j</math> and the matrix of coefficients <math>L_{ij}</math> in the form
	:<math>X_j = \frac{\mu_j-\mu^{\rm eq}_j}{T}; \;\; \frac{d N_i}{d t}=\sum_j L_{ij}X_j </math>		:<math>X_j = \frac{\mu_j-\mu^{\rm eq}_j}{T}; \;\; \frac{d N_i}{d t}=\sum_j L_{ij}X_j </math>

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	:<math>L_{ij}=-\frac{V}{R}\sum_r w^{\rm eq}_r \gamma_{ri}\gamma_{rj}</math>		:<math>L_{ij}=-\frac{V}{R}\sum_r w^{\rm eq}_r \gamma_{ri}\gamma_{rj}</math>

	These symmetry relations, <math>L_{ij}=L_{ji}</math>, are exactly the Onsager reciprocal relations. The coefficient matrix <math>L</math> is non-positive. It is negative on the linear span of the stoichiometric vectors <math>\gamma_{r}</math>.		These symmetry relations, <math>L_{ij}=L_{ji}</math>, are exactly the [[Onsager reciprocal relations]]. The coefficient matrix <math>L</math> is non-positive. It is negative on the linear span of the stoichiometric vectors <math>\gamma_{r}</math>.

	So, the Onsager relations follow from the principle of detailed balance in the linear approximation near equilibrium.		So, the Onsager relations follow from the principle of detailed balance in the linear approximation near equilibrium.

← Older revision		Revision as of 16:54, 8 February 2012
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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 isobaric systems with the constant [[enthalpy]].		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 [[Isoenthalpic–isobaric ensemble \|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:
	:<math>\frac{d N_i}{d t}=V\sum_r \gamma_{ri} w_r=V\sum_r (\beta_{ri}-\alpha_{ri})w_r </math>		:<math>\frac{d N_i}{d t}=V\sum_r \gamma_{ri} w_r=V\sum_r (\beta_{ri}-\alpha_{ri})w_r </math>