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Comment on the kinetic potential and the maxwell construction in non-equilibrium chemical phase transitions
Volume 62A, number 7
COMMENT
PHYSICS LETTERS
ON THE KINETIC POTENTIAL NON-EQUILIBRIUM
3 October 1977
AND THE MAXWELL
CHEMICAL
CONSTRUCTION
IN
PHASE TRANSITIONS
G. NICOLIS and R. LEFEVER Facult( des Sciences de l'Universitd Libre de Bruxelles, Campus Plaine, 1050 Bruxelles, Belgium Received 22 June 1977 A chemical model involving nonequilibrium transitions is considered. The Landau-Ginzburg like potential determining the steady-state probability distribution is shown to contain odd terms in the order parameter. It is concluded that the location of the transition point in parameter space cannot be inferred from the deterministic rate equations. Several chemical models are known which, under spatially homogeneous conditions, predict abrupt transitions between simultaneously stable steady states. For a well-defined choice of parameter values the distance between these states tends to zero, and a transition similar to a "second order" transition involving a critical point is predicted. The simplest system illustrating this behavior is the Schl6gl model [1] kl k3 A+2X~ 3X, X~- B, (1) k2 k4 where X denotes the variable intermediate, and the concentration of A, B is supposed to be controlled externally. Introducing two parameters 6 , 6 ' through k3=3+6,
(2)
BM = 1+8',
Fig. 1. x =0
for all 8 ,
(6a)
in addition to two real roots x.=+X/L-ff
for
8<0.
(6b)
we obtain a cubic equation determining the steady-state values of X, Under the additional transformation
The analogy with Van der Waals theory has prompted several authors [ 1 - 4 ] to define the transition point in the multiple steady-state region in terms o f a Maxwell construction based on the properties of the macroscopic rate equation alone. In the particular case o f the Schl6gl model the latter takes the form
X = A(1 + x ) ,
(4)
dx/d t = - 3 V/Ox ,
(5)
where the potential function V is easily found to be (up to an arbitrary constant)
and fixing all others to the values .1 k 2 = l / A 2,
k 1 = 3 / A 2,
k4=1,
(3)
this equation becomes 6'-6
=x 3+6x.
Fig. 1 describes the unfolding o f the cubic in the parameter space. At the cusp singularity, 6 = 3 ' = O, one has the triple root x = O, i.e., X = A . Along the line 8 = 6' one has a root ,1 We assume a fixed density of A. Thus, the volume dependence of the rate constants can also be expressed through their dependence of the number of particles of A.
x4 x2 V= T + 8 T +
(8 - 8 ' ) x .
(7)
(8)
This relation predicts that the transition line - the "line of conflict" in the therminology of catastrophe theory [3] - is 8 = 8'
(9)
Indeed, along this line Vhas one maximum and two 469
Volume 62A, number 7
PHYSICS LETTERS
equal height minima V(N/r~) = V ( - x / - ~ ) = ~ +
522 - 3524 '
(10)
Note the similarity between (8) and the L a n d a u Ginzburg Hamiltonian. In particular, were the "external field" term (5 - 6 ' ) x absent, V would be completely symmetrical in the "order parameter" x. Recently, a solution to the stochastic analog of the Schl6gl model was constructed [ 5 - 8 ] . In the thermodynamic limit, A ~ ~, the stationaiy probability distribution corresponding to a birth-and-death process is
P ( X ) cx exp [A U(X )] ,
(11)
where the dominant contributions to U are: X XInX+XIn3+XIn U(X) =-~ - --~
(X/A)2+k 2 (X/A) 2 + ?t2
+ 2 k a r c t g ( X / A k ) - 2Xarctg(X/AX) ,
(12)
with k2=o+O
,
•
(13)
Expanding eq. (12) around X = A (cf. eq. (4)) one obtains, for small 5 and 5': U =8"~-8 x + 5 ~~ 3 5 ' x 2 +~-X3 5' -- @6x4 + O(x5,52, 5'2,55').
(14)
This is different from eq. (8). In particular, the transition line is not the symmetry axis of the cuspoid, 8 = 8'. This implies that the stationary probability distribution is asymmetrical around the unstable deterministic steady state ,2 or, equivalently, that the coexistence region is not symmetrical around the "critical isochore" x = 0. ,2 U could also be expanded around the stochastic mean. One can verify that such an expansion involving a redefined order parameter has the same properties as eq. (14). Moreover, one can show that the third order variance, ((8 X) 3) of the stationary probability distribution is non-vanishing and of order A 3. This establishes in an alternative way the skewness of the probability distrNution. We are indebted to M. MalekMansour for this remark. The question of finding a new order parameter in terms of which U would become symmetrical is not pursued here. 470
3 October 1977
Surprising as it is, this result is reminiscent o f some suggestions made in the context of liquid-vapour transitions [9] Concerning possible deviations from the law of rectilinear diameters. Moreover the occurrence of odd terms in the Landau-Ginzburg Hamiltonian, which frequently turns out to be irrelevant for critical behavior [10], may well be important in determining the correct location of the first order transition point. This is exactly what happens with eq. (14). Let us comment briefly on this striking breakdown of the deterministic predictions about the transition point. In the birth- and death master equations, the transition probabilities for the four reactions in (1) are monomials respectively of second, third, first and zeroth degree in X. Thus, the master equation has no obvious symmetry built into it. This is particularly clear in the nonlinear Fokker-Planck equation which can be constructed from the master equation [11]. Whereas the friction term has the same parity properties as the deterministic equation, the (nonlinear) diffusion term has entirely different properties. If the effective diffusion coefficient were constant - as it is usually postulated in Fokker-Planck equations based on the use of Langevin forces - the discrepancy would disappear and one would recover the deterministic potential, eq. (8). It is worth noting that the stochastic potential U, eq. (14), can be derived from a thermodynamic argument as follows. In the notation of eq. (2) and (3) the rate equation at the steady state reads:
X 1 X2/A2+3+8 =A. 3 X2/A 2 + (1 + 8' )/3
(15)
This can be interpreted as the equilibrium condition of the following reaction:
L(X) X< ~ D , k
kD=A ,
(16a)
with a cooperative step involving the effective rate constant L ( X ) = 1 (X/A)2+3+8 3 (X/A)2 + (1 + fi ')/3 '
(16b)
D being a reservoir variable controlled from outside. The "Gibbs free energy" for the reactive system, eq. (16a) obeys to the relation: d(~= ~0jcd X ,
(16c)
Volume 62A, number 7
PHYSICS LETTERS
where, in addition to the perfect gas contribution ~0X cx In X, the "chemical potential" ~0X comprises a part related to the "activity coefficient" L ( X ) , eq. (16b). This yields 1 dG - l n 3 + l n ~ + l n A kB T d X --
(X/A)2+3+~ ( X / A ) 2 + (I + 5 ' )/3'
(17) where k B is Boltzmann's constant and T is the temperature. The right hand side is the negative of the differential of the stochastic potential U, eq. (12). We conclude that: U = - (1/k B T ) G + const.
3 October 1977
We conclude that the probability distribution in the vicinity of a nonequilibrium phase transition as well as the location of the transition point in parameter space cannot be inferred from arguments based exclusively on the deterministic rate equations. This illustrates the type o f qualitatively new insight obtained by stochastic analysis of nonequilibrium systems. We have greatly benefited from stimulating discussions with I. Prigogine, M. Malek-Mansour, J.W. Turner and W. Horsthemke.
(I 8)
This relation, along with (16c), enables one to express the transition between multiple steady states in terms of a Maxwell construction based on the equality o f the values o f the "chemical potential" ~0j( on the two shnultaneously stable branches. We believe that the validity of detailed balance condition in the master equation for system (1) is responsible for this remarkable relation between stochastic and "thermodynamic" potentials. On the other hand, if the Gibbs free energy G of the nonequilibrium system (1) is computed directly using a local equilibrium assumption, the result turns out to be completely different both from G or U and from the deterministic potential V. Different results are also obtained by integrating the differential form dxPoccurring in the Glansdorff-Prigogine evolution criterion [12], although in the latter case one does find o d d t e r m s in x in the resulting potential.
References [1] F. Schl6gl, Z. Physik 248 (1971) 446. [2] Y. Kobatake, Physica 48 (1970) 301. [3] R. Thorn, Stabilit~ structurelle et morphog6n~se (Benjamin, New York, 1972). [4] D. Bedaux, P. Mazur and R.A. Pasmanter, Physica 86A (1977) 355. [5] H.K. Janssen, Z. Physik 270 (1974) 67. [6] K. Kitahara, Ph. D. Dissertation, Univ. of Brussels (1974) [7] I.S. Matheson, D.F. Walls and C.W. Gardiner, J. Stat. Phys. 12 (1975) 21. [8] G. Nicolis and J.W. Turner, Physica A, in press. [9] J.J. Rahr and N.D. Mermin, Phys. Rev. A8 (1973) 472. [10] B. Halperin, P. Hohenberg and E. Siggia, Phys. Rev. B13 (1976) 1229. [11] W. Horsthemke and L. Brenig, Z. Phys. B, in press. [12] P. Glansdorff and I. Prigogine, Thermodynamic theory of structure, stability and fluctuations (Wiley, New York, 1971).
47/
COMMENT
PHYSICS LETTERS
ON THE KINETIC POTENTIAL NON-EQUILIBRIUM
3 October 1977
AND THE MAXWELL
CHEMICAL
CONSTRUCTION
IN
PHASE TRANSITIONS
G. NICOLIS and R. LEFEVER Facult( des Sciences de l'Universitd Libre de Bruxelles, Campus Plaine, 1050 Bruxelles, Belgium Received 22 June 1977 A chemical model involving nonequilibrium transitions is considered. The Landau-Ginzburg like potential determining the steady-state probability distribution is shown to contain odd terms in the order parameter. It is concluded that the location of the transition point in parameter space cannot be inferred from the deterministic rate equations. Several chemical models are known which, under spatially homogeneous conditions, predict abrupt transitions between simultaneously stable steady states. For a well-defined choice of parameter values the distance between these states tends to zero, and a transition similar to a "second order" transition involving a critical point is predicted. The simplest system illustrating this behavior is the Schl6gl model [1] kl k3 A+2X~ 3X, X~- B, (1) k2 k4 where X denotes the variable intermediate, and the concentration of A, B is supposed to be controlled externally. Introducing two parameters 6 , 6 ' through k3=3+6,
(2)
BM = 1+8',
Fig. 1. x =0
for all 8 ,
(6a)
in addition to two real roots x.=+X/L-ff
for
8<0.
(6b)
we obtain a cubic equation determining the steady-state values of X, Under the additional transformation
The analogy with Van der Waals theory has prompted several authors [ 1 - 4 ] to define the transition point in the multiple steady-state region in terms o f a Maxwell construction based on the properties of the macroscopic rate equation alone. In the particular case o f the Schl6gl model the latter takes the form
X = A(1 + x ) ,
(4)
dx/d t = - 3 V/Ox ,
(5)
where the potential function V is easily found to be (up to an arbitrary constant)
and fixing all others to the values .1 k 2 = l / A 2,
k 1 = 3 / A 2,
k4=1,
(3)
this equation becomes 6'-6
=x 3+6x.
Fig. 1 describes the unfolding o f the cubic in the parameter space. At the cusp singularity, 6 = 3 ' = O, one has the triple root x = O, i.e., X = A . Along the line 8 = 6' one has a root ,1 We assume a fixed density of A. Thus, the volume dependence of the rate constants can also be expressed through their dependence of the number of particles of A.
x4 x2 V= T + 8 T +
(8 - 8 ' ) x .
(7)
(8)
This relation predicts that the transition line - the "line of conflict" in the therminology of catastrophe theory [3] - is 8 = 8'
(9)
Indeed, along this line Vhas one maximum and two 469
Volume 62A, number 7
PHYSICS LETTERS
equal height minima V(N/r~) = V ( - x / - ~ ) = ~ +
522 - 3524 '
(10)
Note the similarity between (8) and the L a n d a u Ginzburg Hamiltonian. In particular, were the "external field" term (5 - 6 ' ) x absent, V would be completely symmetrical in the "order parameter" x. Recently, a solution to the stochastic analog of the Schl6gl model was constructed [ 5 - 8 ] . In the thermodynamic limit, A ~ ~, the stationaiy probability distribution corresponding to a birth-and-death process is
P ( X ) cx exp [A U(X )] ,
(11)
where the dominant contributions to U are: X XInX+XIn3+XIn U(X) =-~ - --~
(X/A)2+k 2 (X/A) 2 + ?t2
+ 2 k a r c t g ( X / A k ) - 2Xarctg(X/AX) ,
(12)
with k2=o+O
,
•
(13)
Expanding eq. (12) around X = A (cf. eq. (4)) one obtains, for small 5 and 5': U =8"~-8 x + 5 ~~ 3 5 ' x 2 +~-X3 5' -- @6x4 + O(x5,52, 5'2,55').
(14)
This is different from eq. (8). In particular, the transition line is not the symmetry axis of the cuspoid, 8 = 8'. This implies that the stationary probability distribution is asymmetrical around the unstable deterministic steady state ,2 or, equivalently, that the coexistence region is not symmetrical around the "critical isochore" x = 0. ,2 U could also be expanded around the stochastic mean. One can verify that such an expansion involving a redefined order parameter has the same properties as eq. (14). Moreover, one can show that the third order variance, ((8 X) 3) of the stationary probability distribution is non-vanishing and of order A 3. This establishes in an alternative way the skewness of the probability distrNution. We are indebted to M. MalekMansour for this remark. The question of finding a new order parameter in terms of which U would become symmetrical is not pursued here. 470
3 October 1977
Surprising as it is, this result is reminiscent o f some suggestions made in the context of liquid-vapour transitions [9] Concerning possible deviations from the law of rectilinear diameters. Moreover the occurrence of odd terms in the Landau-Ginzburg Hamiltonian, which frequently turns out to be irrelevant for critical behavior [10], may well be important in determining the correct location of the first order transition point. This is exactly what happens with eq. (14). Let us comment briefly on this striking breakdown of the deterministic predictions about the transition point. In the birth- and death master equations, the transition probabilities for the four reactions in (1) are monomials respectively of second, third, first and zeroth degree in X. Thus, the master equation has no obvious symmetry built into it. This is particularly clear in the nonlinear Fokker-Planck equation which can be constructed from the master equation [11]. Whereas the friction term has the same parity properties as the deterministic equation, the (nonlinear) diffusion term has entirely different properties. If the effective diffusion coefficient were constant - as it is usually postulated in Fokker-Planck equations based on the use of Langevin forces - the discrepancy would disappear and one would recover the deterministic potential, eq. (8). It is worth noting that the stochastic potential U, eq. (14), can be derived from a thermodynamic argument as follows. In the notation of eq. (2) and (3) the rate equation at the steady state reads:
X 1 X2/A2+3+8 =A. 3 X2/A 2 + (1 + 8' )/3
(15)
This can be interpreted as the equilibrium condition of the following reaction:
L(X) X< ~ D , k
kD=A ,
(16a)
with a cooperative step involving the effective rate constant L ( X ) = 1 (X/A)2+3+8 3 (X/A)2 + (1 + fi ')/3 '
(16b)
D being a reservoir variable controlled from outside. The "Gibbs free energy" for the reactive system, eq. (16a) obeys to the relation: d(~= ~0jcd X ,
(16c)
Volume 62A, number 7
PHYSICS LETTERS
where, in addition to the perfect gas contribution ~0X cx In X, the "chemical potential" ~0X comprises a part related to the "activity coefficient" L ( X ) , eq. (16b). This yields 1 dG - l n 3 + l n ~ + l n A kB T d X --
(X/A)2+3+~ ( X / A ) 2 + (I + 5 ' )/3'
(17) where k B is Boltzmann's constant and T is the temperature. The right hand side is the negative of the differential of the stochastic potential U, eq. (12). We conclude that: U = - (1/k B T ) G + const.
3 October 1977
We conclude that the probability distribution in the vicinity of a nonequilibrium phase transition as well as the location of the transition point in parameter space cannot be inferred from arguments based exclusively on the deterministic rate equations. This illustrates the type o f qualitatively new insight obtained by stochastic analysis of nonequilibrium systems. We have greatly benefited from stimulating discussions with I. Prigogine, M. Malek-Mansour, J.W. Turner and W. Horsthemke.
(I 8)
This relation, along with (16c), enables one to express the transition between multiple steady states in terms of a Maxwell construction based on the equality o f the values o f the "chemical potential" ~0j( on the two shnultaneously stable branches. We believe that the validity of detailed balance condition in the master equation for system (1) is responsible for this remarkable relation between stochastic and "thermodynamic" potentials. On the other hand, if the Gibbs free energy G of the nonequilibrium system (1) is computed directly using a local equilibrium assumption, the result turns out to be completely different both from G or U and from the deterministic potential V. Different results are also obtained by integrating the differential form dxPoccurring in the Glansdorff-Prigogine evolution criterion [12], although in the latter case one does find o d d t e r m s in x in the resulting potential.
References [1] F. Schl6gl, Z. Physik 248 (1971) 446. [2] Y. Kobatake, Physica 48 (1970) 301. [3] R. Thorn, Stabilit~ structurelle et morphog6n~se (Benjamin, New York, 1972). [4] D. Bedaux, P. Mazur and R.A. Pasmanter, Physica 86A (1977) 355. [5] H.K. Janssen, Z. Physik 270 (1974) 67. [6] K. Kitahara, Ph. D. Dissertation, Univ. of Brussels (1974) [7] I.S. Matheson, D.F. Walls and C.W. Gardiner, J. Stat. Phys. 12 (1975) 21. [8] G. Nicolis and J.W. Turner, Physica A, in press. [9] J.J. Rahr and N.D. Mermin, Phys. Rev. A8 (1973) 472. [10] B. Halperin, P. Hohenberg and E. Siggia, Phys. Rev. B13 (1976) 1229. [11] W. Horsthemke and L. Brenig, Z. Phys. B, in press. [12] P. Glansdorff and I. Prigogine, Thermodynamic theory of structure, stability and fluctuations (Wiley, New York, 1971).
47/