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The phase transition and classification of critical points in the multistability chemical reactions
36
Communications
in Nonlinear
Science & Numerical
Simulation
March 2000
The phase transition and classification of critical points in the multistability chemical reactions Chunhua ZHANG* , Fugen WU’, *Department **Department (Received
of Physics, of Applied
January
Chunyan WU* and Fa OU**
Guangdong University Physics, South China
of Technology, Guangzhou 510090, China University of Technology, Guangzhou 510641,
China
17, 2000)
Abstract: In this paper, we study the phase transition and classification of critical points in multistability chemical reaction systems. Referring to the spirit of Landau’s theory of phase transitions, this paper deals with the varied transitions and critical phenomena in multistable chemical systems. It is demonstrated that the higher the order of the multistability, the wider the variety of phase transitions will be. A classification scheme of critical points according to the stability criterion and the thermodynamic potential continuity is suggested. It is useful for us to study critical phenomena especially in the non-equilibrium systems including the multi-stable chemical ones. Keywords: nonlinear chemistry, non-equilibrium phase transition, critical points
Introduction The multistability in chemical reaction has been paid much attention to at all times[1-31. As chemical multistability acts at the non-linear area, and its critical phenomenon is part of non-equilibrium statistical mechanics of non-linearity, we usually take the critical phenomena and phase transition in the equilibrium thermodynamic for reference when we discuss the critical phenomena and phase transition in non-linear systems L4w61.As we all know, Ehrenfest (see, for example, [7]) d iscovered the conventional classification of phase transition in equilibrium thermodynamics. Its classification symbol is the continuity of potential function and the differential coefficient in thermodynamics. (It can be briefly called &h-order phase transitions when phase transitions of discontinuity happen at nth-order differential coefficient). So far, no other phase transitions of equilibrium have been found except for the above two. When phase transition of non-equilibrium is being discussed, it is only limited to the contrast of first-order and second-order phase transition of equilibrium i4-% s-121. The rich variety of the phase transition of equilibrium is usually overlooked. Additionally, in the equilibrium phase transitions, only second-order phase transition is related with critical point; the first-order phase transition occurs at the equal chemical potential function points of the two phases. Because of this, the critical phenomenon mentioned in equilibrium thermodynamics theory is considered as second-order phase transition (or continuous phase transition). But varieties of critical points exist in the chemical multistability, and the phase transitions connecting with them cannot be simply contrasted with second-order phase transitions of equilibrium. In order to describe the rich varieties of critical phenomena in the multistability chemical reaction, how to classify these critical points becomes the key point of the problem. With reference to the basic theory of Landau [131 phase transitions the critical points are classified in this paper based on the continuity and stability of critical points of thermodynamic function. We also review the variety of the critical points in autocatalytic reactions with
multistability.
Vol. 5, No. 1
1 Dynamic classification
ZHANG
equation, of critical
et al.: Critical points in the multistability
quasi-thermodynamic points
chemical reactions
potential
function
37
and
Assume in an opened chemical reaction, only one ingredient’s concentration is changed; the concentrations of the remaining ingredients are constant as to be under environmental control. Besides, if the reaction can always be kept uniform (as to be irritated), dynamic equation in chemical reaction can be expressed as
where x is the concentration of the changeable ingredient, X is a conditional parameter (maybe one or one set) controlled by environment, F(x, X) is a continuous differentiable function. This equation is an autonomic dynamic equation, so the system has quasi-thermodynamic potential function G(z, X). The quasi-thermodynamic potential function G(z, X) can be determined by the following equationlsl : dx -- r3G z= ax When dx/dt = 0, the static equation of this system is obtained as follows: F(x,,X)
= 0
where the subscript “s” indicates the static state of x. Static state of the system is determined by a set of control parameters. If one set of control parameters has multiple static values, it means that the system is multi-stable. In the tristability system, there are three static state solutions xs . If (a2G/ax2)Z. > 0, the solution is stable; if (a”G/a~~)~, < 0, the solution is unstable. As for the three solutions in the stable section, the solutions of up Xh and down x1 are stable, and the solution of the middle x, is not stable (zh < x, < x1). If (a2G/0x2)z. = 0, there are two solutions of fringe critical points: xt (corresponding to the upper threshold point) and XJ (corresponding to the lower threshold point), which are not stable. Moreover, when the control parameter is adjusted to have the transition from bistability to single-stability, the system is to be another kind of critical point (termination point of bistability), the equation is as follows18~ ‘I: (g,,
= (g),
=0
and
(g),
> 0
0)
This critical point is a stable inflexion, where K is critical point. Both of the above critical points can be analogized with first-order and second-order phase transitions in equilibrium. By further analysis, it is found that, in chemical reaction with five or more stabilities, there are not only the above two critical points which can be analogized with first-order and second-order phase transitions in equilibrium, but also critical points which are not in equilibrium phase transition. The critical points are necessarily classified as follows, so that the variety of critical points in chemical multistability is clearly explained. A critical point K is called first-kind critical point if =0
and
First-kind critical points are unstable constantly L81. The fringe critical points in tristability system or above belong to this kind of critical points. Furthermore, a critical point K is called second-kind critical point if conditions (1) hold. In this case, K is stable. The critical point of the bistability terminal point, as we discussed in paper [8], belong to this kind. Suppose (g,,
= (s),
= 0
and
(g),
< 0
38
Communications
in Nonlinear
Science & Numerical
March 2000
Simulation
then this critical point K is called sub-second-kind critical point, and it is unstable. Generalized from the above, a critical point K is called odd-&h-kind critical point if
a2G
(@- >K=”
.=(g),=O
and
(s),#O,
n=l,3,5;..
(2)
It is not stable for ever. If
.=
(=),=(I
and
iEi:z:
n = 2,4,6,.
..
7 then we say that the former critical points are sub-even-nth-kind critical points, and they are also unstable; the latter critical points are even-&h-kind critical points, and they are stable. The critical points as defined above can probably exist in multistability system. The more the stability branches, the more categories of critical points will exist, and the higher the category order will be. We will take the following tristability chemical reaction model as an example, to explore its phase transition and critical phenomena.
2 The phase transition reaction
and the critical
point in the tristability
chemical
Schlogl151 has raised the following autocatalytic reaction model which has only one middle ingredient variable, and not including diffusion process: B$X,
A+2Xs3X
where /CO,ICI, Icg, ks are rate constants of the reaction. Assume the reaction process does not involve heat-effect; all the ingredients make up the ideal solution; and the reaction dynamics satisfy the Mass action law. When the reaction conditions (temperature, pressure, etc.) are fixed, the entire rate coefficient is constant. Suppose only A and B can be exchanged with the environment, and by doing so maintain their constant concentration in the system, while ingredient X cannot be exchanged with the environment, and its concentration depends completely on dynamic property of the system. Therefore, the following rate equation is obtained:
-dx = +x3 dt
+ k2az2 - LIZ + kob
where x indicates the concentration of the ingredient X. This rate equation is based on the chemical disorder impact concept. This concept agrees that the impact probability between the molecules is equal, the reaction rate is in direct ratio to frequency of the reaction molecule impact, and in direct ratio to their concentration. This case can only exist in the so-called ideal solution. In the non-ideal solution, the molecule movement cannot be in complete disorder due to the different molecule structure and different reciprocity, and therefore the molecule impact cannot be in complete equal probability. The reaction rate cannot be simply in direct ratio to the solution concentration. In addition, considering the noise influence to the temperature and pressure, the rate coefficient probably has something to do with the concentration. As paper [2] said, in the ideal conditions, the maximum uniformity stability solutions to the above Schlogl model is three; and in the non-ideal conditions, the relative stability solutions can reach six in
Vol. 5, No. 1
ZHANG
et al.: Critical
the range of some parameters. Therefore, have five stability solutions maximally: da:
points in the multistability
39
we assume the following rate equation, which may dG -y&
-x5+px3+qx+r=
dt=
chemical reactions
Here, the second equation means the existence of the potential function G; while p, q and r are all plus or minus real numbers, which are dependent on the reaction rate. 2.1
Static
solutions
F’rom dx/dt
and
its stability
= 0, we obtain the static equation as r=xz-px,3-qxs
(3)
Firstly we will review the static solutions when P = 0, as it is valuable to help understand the multistability around r = 0. When r = 0, x5 has five roots, as follows xs=o,
*Jz
*t
P*=;(P+@T-q
And the conditions for real roots are: p>o, The stability
q
p2>-4q
of the five roots are decided by the sign of (82G/8x2)5=2S because d2G dr == 5x; - 3px,2 - q c-1 8x2 z=z. dx,
Evidently, (62G/dx2),e=s = -q > 0, therefore the points (r = 0, x, = 0) are stable, as well as a2G (-1dX2 h=*&
+/E
= +*GG
So, it is stable at the point of r = 0, x8 = f fi, while it is unstable at the point of T = 0, x, = f fl. The following is concluded thereby. Along the static curve on the x, N T area, there are three stable branches, the intercepts of which are respectively 6, 0, --fi on the xS axis, and are respectively called “High, Midst or Low” branch; there are also two unstable branches, the intercepts of which are f@, as shown in Fig. 1. In this case, the system is tristability. 2.2
First-kind
critical
-15
0
r Fig. 1 The curve of concentration zS and parameter T (p = 6, q = -6)
point
From a2G = 5x; - 3px,2 - q = 0 c-1 8x2 5=2.
40
Communications
first-kind
in Nonlinear
Science & Numerical
Simulation
March 2000
critical point is concluded: xs=&T,
-6;
a* = $(bf
J9m)
(4
Evidently, the conditions of four real roots are as follows: 9p2+2oq
> 0,
q < 0,
p > 0
(5)
-6, -,/GZ, $57, & are the x, values of Low branch up-jump, Midst branch downjump, Midst branch up-jump and High branch down-jump respectively (refer to Fig. 1):
-6, “t(-1 zzz
XI”’ = -&ir,
xy = @j-, xp’ = 6
(6)
where the superscripts (-), (0) and (+) are indicate Low, Midst and High branch respectively. According to Eq. (3), the “P values corresponding to the four critical points are as follows: Tt(--I = 7-(-&q,
ry’ = T-(-&r),
Ty) = 7-(&q,
ry’
= r(&q
From Eq. (2), we have
X=+/Q
= 20x,3 - 6px, = 2x,(10x;
- 3~) = 2&*(9p2
+ 20q) # 0
Therefore, all the four critical points are unsta2.5 ble, and belong to first-kind critical point, indicating the transition of High, Midst and Low stable branch respectively (phase transition). l+rther more, when p” > -4q, p2 + -4q, the tristable area disappear gradually; when p2 < -4q (but p2 > -2Oq/9), the tristable ?T 0 area disappear completely, but the High, Midst and Low stable branches still exist, forming two non-overlap bistable curve in the quadrants 1 and 3 of the area of zS - T (see Fig. 2). The two non-stackable bistable curve are called two-2.5 step bistability. When p2 = -4q we have -15 0 (+I = r$-’ = 0. This critical status means the r TJ transition between the tristability and the twoFig. 2 The curve of concentration zS and step bistability. But the critical points correparameter T (p = 6, q = -9) sponding to the above transition are the lower threshold point of the High Branch and the upper threshold point of the Low Branch, but they still belong to first-kind critical point. 2.3
Second-kind
critical
I
point
When p2 -+ -2Oq/9, we can see, from the equations of (4)-(6), that the upper and lower threshold points on the two two-step bistable curves are close to each other. We have the following limit conditions:
ZHANG
Vol. 5. No. 1
et
ol.:
Critical Doints in the multistabilitv
chemical reactions
41
This means that each of the two bistable curves is condensed to one point, becoming the inflexion of the curve xS N r, and is located in the quadrants 1 and 3 respectively (see Fig. 3). The reason is as follows: d3r d3G = (-1 dxz +.=*@g ax3 +-y,Z.=*~fi a4G = 12p > 0 C--J ax4 p”=-E.p,z.=f;fl
(3
=o
Therefore, this inflexion is second-kind (stable) critical point, corresponding to the transition between two-step bistability and single stability. When p > 0, q < 0, p’ > -49, firstly keep p unchanged and q approaching to 0, in this case, the Midst stable branch will be gradually shortened. Under the limit conditions lim xy’ = fle &C q+o
= 0,
lim x:0) = liis(-@C) P-+0
= 0
the Midst branch is condensed to one point, that is the origin of the area of x, N r (x, = 0, r = 0). We have d2r a3G ax3 q=o,z.=o = (3dx;
(-1
this point becomes an inflexion, but becomes the connecting point of the The sub-second-kind critical point (r and the single bistability (or one-step
q=o, z.=o
b&able loops and completely singlestable curve (p = 3, q = -81/20) Fourth-kind
critical
a4G = -6p < 0 ( __ a24 1 q=o, 2.=0
belongs to sub-second-kind (unstable) critical point. It original upper and lower unstable branches (see Fig. 4). = 0, x, = 0) indicates the transition between tristability bistability).
Fig. 3 The transition between two-step
2.4
= 0,
Fig. 4 The transition between tristability and bistability (p = 3, q = 0)
points
When q 1 0, p2 > -49, there is always a single bistable curve as shown in Fig. 5 (when q > 0, the origin xS = 0, r = 0 are no longer critical points), the intercepts of High and Low branches on the xs axis are 6, -fl respectively. The threshold points on the left of the
42
Communications
in Nonlinear
Science & Numerical
Simulation
March 2000
High branch and the right of the low branch are zr) = &, xi-1 = -& respectively. The decrease of ] Q ] and p2 will shorten the area of the b&able curve, under the following limit conditions: lim XI+’ = lilie z$-’ = ;$ (+G) = ;lo wm = 0 (I’0 $-to
p2+o
p*-+0
p2-to
They are condensing to one point. At this time, the origin is back to the critical point, indicating the transition between the bistability and the single stability. As and the critical point K is fourth-kind (stable) critical point, which is the highest order of critical point in the tristability system.
(-;;)K
= 120 > 0
*.’
3 Remark K Based on the above classification of critig 0 cal points in the multistability chemical reaction system, we discussed the phase transition and the critical points in the tristability autocatalytic reaction under the non-ideal conditions, -2.0 and revealed the variety of the phase transition -25 0 and the critical points in the multistability chemr ical reaction system sufficiently. With the same method, we demonstrated that the higher the orFig. 5 The transition between two-step der of the multistability, the more complex and b&ability and single stability abundant the critical points are. cp = 0, q = 0) Both equilibrium phase transition and nonequilibrium phase transition are the break of some physical variables happening in the system, and are usually accompanied by the sequence change. But the mechanisms of phase transitions are different. The equilibrium phase transition comes from the reciprocity between the molecules; the non-equilibrium phase transition depends on the macro-kinetics of the non-equilibrium. The content of the non-equilibrium is much abroad, since the nonlinear non-equilibrium system has much more movement format than the equilibrium system. In this paper, the multistability chemical reaction phase transition and the classification of the critical points are the reflection of this phenomenon. Like the equilibrium phase transition, there lies some abnormal physics-chemical characteristics around the critical points of the non-equilibrium phase transition, and they are much more abundant. To understand and apply the abnormal characteristics around the variety of critical points will significantly improve the security and stability especially in chemical industry.
References [l] Vidal, C. and Pacauit, A., Nonlinear Phenomena in Chemical Dynamics, Springer-Verlag, Berlin, 1981 [2] Li Rusheng, Non-equilibrium Thermodynamics and Dissipative Structures, Tsinghua Univ. Press; Beijing, 1996 (in Chinese)
[3] Gao Qingyu, Cai Zunsheng and Zhao Xuezhuang, Nonlinear chemical reaction dynamics, Progress in Chemistry,
1997, 9(l):
59 (in Chinese)
Vol. 5, No. 1
GUO et al.: Blow up problem for Landau-Lifshitz
43
equations in two dimensions
[4] Li Rusheng, Equilibrium and Non-equilibrium Statistical Mechanics, Tsinghua Univ. Press, Beijing, 1995 (in Chinese) [5] Wu Fugen, Zhan Yehong and Wu Tingwan et al., The fluctuation of bistability in chemical reactions, Acta Physico-Chimica Sinica, 1998, 14(7): 659-663 [6] Zhan Yehong, Wu Fugen and Ou Fa, Critical phenomena of chemical reactions and its secondorder-like transition character, Chinese J. Chem. Phys., 1999, 12(l): 57-62 [7] Xu Zuyao, The Principle of Phase Transition, Science, 1988 (in Chinese) [8] Ou Fa and Wu Fugen, Critical phenomena of the bistability in chemical reactions and Landau theory of phase transitions, Acta Chimica Sinica, 1996, 54: 218 (in Chinese) [9] Wu Fugen, Zhan Yehong and Wu Tingwan et al., The slowing down in the bistability chemical reaction, J. Molecular Sci., 1998, 121(4): 199-203 [lo] Hu Gang, Stochastic Forces and Nonlinear Systems, Shanghai Scientific and Technological Education Press, Shanghai, 1994 (in Chinese) [ll] Xin Houwen, Reaction Kinetics in Fractal Media, Shanghai Scientific and Technological Education Press, Shanghai 1997 (in Chinese) [12] Reichl, L. E., A Modern Course in Statistical Physics, Univ. of Texas Press, Texas, 1980 [13] Landau, L. D. and Lifshitz, E. M., Statistical Physics, Pergamon, Oxford, 1958
Blow up problem for Landau-Lifshitz in two dimensions
equations
Boling GUO’, Yongqian HAN* and Ganshan YANG** *Institute of Applied Physics and Computational Mathematics, Nonlinear Center for Studies, P. 0. Box 8009, Beijing 100088, China ** Graduate School of China Academy of Engineering Physics, P. 0. Box 2101, Beijing 100088, (Received
March
China
7, 2000)
Abstract: The solutions of two dimensional Landau-Lifshitz equations, finite time, are obtained. Keywords: blow up, two dimensional Landau-Lifshitz equations
which blow up in
In 1986, Zhou and Guo in [l] proved the global existence of weak solution for generalized Landau-Lifshitz equations without Gilbert term in multi-dimensions. They consider the homogeneous boundary problem Z(x,t)
= 0,
for x E dR, 0 5 t 5 T
(1)
with the initial value condition
Z(x,O) = $4x), for the system of ferromagnetic
for x E !2
(2)
chain with several variables Zt = 2 x AZ + f(x,
t, Z)
(3)
where f(x, t, Z) is a given S-dimensional vector function in x E Rn, t E R+, Z E R3, p(x) is a given 3-dimensional initial value function on 0, R is a bounded domain in n-dimensional Euclidean space Rn.
Communications
in Nonlinear
Science & Numerical
Simulation
March 2000
The phase transition and classification of critical points in the multistability chemical reactions Chunhua ZHANG* , Fugen WU’, *Department **Department (Received
of Physics, of Applied
January
Chunyan WU* and Fa OU**
Guangdong University Physics, South China
of Technology, Guangzhou 510090, China University of Technology, Guangzhou 510641,
China
17, 2000)
Abstract: In this paper, we study the phase transition and classification of critical points in multistability chemical reaction systems. Referring to the spirit of Landau’s theory of phase transitions, this paper deals with the varied transitions and critical phenomena in multistable chemical systems. It is demonstrated that the higher the order of the multistability, the wider the variety of phase transitions will be. A classification scheme of critical points according to the stability criterion and the thermodynamic potential continuity is suggested. It is useful for us to study critical phenomena especially in the non-equilibrium systems including the multi-stable chemical ones. Keywords: nonlinear chemistry, non-equilibrium phase transition, critical points
Introduction The multistability in chemical reaction has been paid much attention to at all times[1-31. As chemical multistability acts at the non-linear area, and its critical phenomenon is part of non-equilibrium statistical mechanics of non-linearity, we usually take the critical phenomena and phase transition in the equilibrium thermodynamic for reference when we discuss the critical phenomena and phase transition in non-linear systems L4w61.As we all know, Ehrenfest (see, for example, [7]) d iscovered the conventional classification of phase transition in equilibrium thermodynamics. Its classification symbol is the continuity of potential function and the differential coefficient in thermodynamics. (It can be briefly called &h-order phase transitions when phase transitions of discontinuity happen at nth-order differential coefficient). So far, no other phase transitions of equilibrium have been found except for the above two. When phase transition of non-equilibrium is being discussed, it is only limited to the contrast of first-order and second-order phase transition of equilibrium i4-% s-121. The rich variety of the phase transition of equilibrium is usually overlooked. Additionally, in the equilibrium phase transitions, only second-order phase transition is related with critical point; the first-order phase transition occurs at the equal chemical potential function points of the two phases. Because of this, the critical phenomenon mentioned in equilibrium thermodynamics theory is considered as second-order phase transition (or continuous phase transition). But varieties of critical points exist in the chemical multistability, and the phase transitions connecting with them cannot be simply contrasted with second-order phase transitions of equilibrium. In order to describe the rich varieties of critical phenomena in the multistability chemical reaction, how to classify these critical points becomes the key point of the problem. With reference to the basic theory of Landau [131 phase transitions the critical points are classified in this paper based on the continuity and stability of critical points of thermodynamic function. We also review the variety of the critical points in autocatalytic reactions with
multistability.
Vol. 5, No. 1
1 Dynamic classification
ZHANG
equation, of critical
et al.: Critical points in the multistability
quasi-thermodynamic points
chemical reactions
potential
function
37
and
Assume in an opened chemical reaction, only one ingredient’s concentration is changed; the concentrations of the remaining ingredients are constant as to be under environmental control. Besides, if the reaction can always be kept uniform (as to be irritated), dynamic equation in chemical reaction can be expressed as
where x is the concentration of the changeable ingredient, X is a conditional parameter (maybe one or one set) controlled by environment, F(x, X) is a continuous differentiable function. This equation is an autonomic dynamic equation, so the system has quasi-thermodynamic potential function G(z, X). The quasi-thermodynamic potential function G(z, X) can be determined by the following equationlsl : dx -- r3G z= ax When dx/dt = 0, the static equation of this system is obtained as follows: F(x,,X)
= 0
where the subscript “s” indicates the static state of x. Static state of the system is determined by a set of control parameters. If one set of control parameters has multiple static values, it means that the system is multi-stable. In the tristability system, there are three static state solutions xs . If (a2G/ax2)Z. > 0, the solution is stable; if (a”G/a~~)~, < 0, the solution is unstable. As for the three solutions in the stable section, the solutions of up Xh and down x1 are stable, and the solution of the middle x, is not stable (zh < x, < x1). If (a2G/0x2)z. = 0, there are two solutions of fringe critical points: xt (corresponding to the upper threshold point) and XJ (corresponding to the lower threshold point), which are not stable. Moreover, when the control parameter is adjusted to have the transition from bistability to single-stability, the system is to be another kind of critical point (termination point of bistability), the equation is as follows18~ ‘I: (g,,
= (g),
=0
and
(g),
> 0
0)
This critical point is a stable inflexion, where K is critical point. Both of the above critical points can be analogized with first-order and second-order phase transitions in equilibrium. By further analysis, it is found that, in chemical reaction with five or more stabilities, there are not only the above two critical points which can be analogized with first-order and second-order phase transitions in equilibrium, but also critical points which are not in equilibrium phase transition. The critical points are necessarily classified as follows, so that the variety of critical points in chemical multistability is clearly explained. A critical point K is called first-kind critical point if =0
and
First-kind critical points are unstable constantly L81. The fringe critical points in tristability system or above belong to this kind of critical points. Furthermore, a critical point K is called second-kind critical point if conditions (1) hold. In this case, K is stable. The critical point of the bistability terminal point, as we discussed in paper [8], belong to this kind. Suppose (g,,
= (s),
= 0
and
(g),
< 0
38
Communications
in Nonlinear
Science & Numerical
March 2000
Simulation
then this critical point K is called sub-second-kind critical point, and it is unstable. Generalized from the above, a critical point K is called odd-&h-kind critical point if
a2G
(@- >K=”
.=(g),=O
and
(s),#O,
n=l,3,5;..
(2)
It is not stable for ever. If
.=
(=),=(I
and
iEi:z:
n = 2,4,6,.
..
7 then we say that the former critical points are sub-even-nth-kind critical points, and they are also unstable; the latter critical points are even-&h-kind critical points, and they are stable. The critical points as defined above can probably exist in multistability system. The more the stability branches, the more categories of critical points will exist, and the higher the category order will be. We will take the following tristability chemical reaction model as an example, to explore its phase transition and critical phenomena.
2 The phase transition reaction
and the critical
point in the tristability
chemical
Schlogl151 has raised the following autocatalytic reaction model which has only one middle ingredient variable, and not including diffusion process: B$X,
A+2Xs3X
where /CO,ICI, Icg, ks are rate constants of the reaction. Assume the reaction process does not involve heat-effect; all the ingredients make up the ideal solution; and the reaction dynamics satisfy the Mass action law. When the reaction conditions (temperature, pressure, etc.) are fixed, the entire rate coefficient is constant. Suppose only A and B can be exchanged with the environment, and by doing so maintain their constant concentration in the system, while ingredient X cannot be exchanged with the environment, and its concentration depends completely on dynamic property of the system. Therefore, the following rate equation is obtained:
-dx = +x3 dt
+ k2az2 - LIZ + kob
where x indicates the concentration of the ingredient X. This rate equation is based on the chemical disorder impact concept. This concept agrees that the impact probability between the molecules is equal, the reaction rate is in direct ratio to frequency of the reaction molecule impact, and in direct ratio to their concentration. This case can only exist in the so-called ideal solution. In the non-ideal solution, the molecule movement cannot be in complete disorder due to the different molecule structure and different reciprocity, and therefore the molecule impact cannot be in complete equal probability. The reaction rate cannot be simply in direct ratio to the solution concentration. In addition, considering the noise influence to the temperature and pressure, the rate coefficient probably has something to do with the concentration. As paper [2] said, in the ideal conditions, the maximum uniformity stability solutions to the above Schlogl model is three; and in the non-ideal conditions, the relative stability solutions can reach six in
Vol. 5, No. 1
ZHANG
et al.: Critical
the range of some parameters. Therefore, have five stability solutions maximally: da:
points in the multistability
39
we assume the following rate equation, which may dG -y&
-x5+px3+qx+r=
dt=
chemical reactions
Here, the second equation means the existence of the potential function G; while p, q and r are all plus or minus real numbers, which are dependent on the reaction rate. 2.1
Static
solutions
F’rom dx/dt
and
its stability
= 0, we obtain the static equation as r=xz-px,3-qxs
(3)
Firstly we will review the static solutions when P = 0, as it is valuable to help understand the multistability around r = 0. When r = 0, x5 has five roots, as follows xs=o,
*Jz
*t
P*=;(P+@T-q
And the conditions for real roots are: p>o, The stability
q
p2>-4q
of the five roots are decided by the sign of (82G/8x2)5=2S because d2G dr == 5x; - 3px,2 - q c-1 8x2 z=z. dx,
Evidently, (62G/dx2),e=s = -q > 0, therefore the points (r = 0, x, = 0) are stable, as well as a2G (-1dX2 h=*&
+/E
= +*GG
So, it is stable at the point of r = 0, x8 = f fi, while it is unstable at the point of T = 0, x, = f fl. The following is concluded thereby. Along the static curve on the x, N T area, there are three stable branches, the intercepts of which are respectively 6, 0, --fi on the xS axis, and are respectively called “High, Midst or Low” branch; there are also two unstable branches, the intercepts of which are f@, as shown in Fig. 1. In this case, the system is tristability. 2.2
First-kind
critical
-15
0
r Fig. 1 The curve of concentration zS and parameter T (p = 6, q = -6)
point
From a2G = 5x; - 3px,2 - q = 0 c-1 8x2 5=2.
40
Communications
first-kind
in Nonlinear
Science & Numerical
Simulation
March 2000
critical point is concluded: xs=&T,
-6;
a* = $(bf
J9m)
(4
Evidently, the conditions of four real roots are as follows: 9p2+2oq
> 0,
q < 0,
p > 0
(5)
-6, -,/GZ, $57, & are the x, values of Low branch up-jump, Midst branch downjump, Midst branch up-jump and High branch down-jump respectively (refer to Fig. 1):
-6, “t(-1 zzz
XI”’ = -&ir,
xy = @j-, xp’ = 6
(6)
where the superscripts (-), (0) and (+) are indicate Low, Midst and High branch respectively. According to Eq. (3), the “P values corresponding to the four critical points are as follows: Tt(--I = 7-(-&q,
ry’ = T-(-&r),
Ty) = 7-(&q,
ry’
= r(&q
From Eq. (2), we have
X=+/Q
= 20x,3 - 6px, = 2x,(10x;
- 3~) = 2&*(9p2
+ 20q) # 0
Therefore, all the four critical points are unsta2.5 ble, and belong to first-kind critical point, indicating the transition of High, Midst and Low stable branch respectively (phase transition). l+rther more, when p” > -4q, p2 + -4q, the tristable area disappear gradually; when p2 < -4q (but p2 > -2Oq/9), the tristable ?T 0 area disappear completely, but the High, Midst and Low stable branches still exist, forming two non-overlap bistable curve in the quadrants 1 and 3 of the area of zS - T (see Fig. 2). The two non-stackable bistable curve are called two-2.5 step bistability. When p2 = -4q we have -15 0 (+I = r$-’ = 0. This critical status means the r TJ transition between the tristability and the twoFig. 2 The curve of concentration zS and step bistability. But the critical points correparameter T (p = 6, q = -9) sponding to the above transition are the lower threshold point of the High Branch and the upper threshold point of the Low Branch, but they still belong to first-kind critical point. 2.3
Second-kind
critical
I
point
When p2 -+ -2Oq/9, we can see, from the equations of (4)-(6), that the upper and lower threshold points on the two two-step bistable curves are close to each other. We have the following limit conditions:
ZHANG
Vol. 5. No. 1
et
ol.:
Critical Doints in the multistabilitv
chemical reactions
41
This means that each of the two bistable curves is condensed to one point, becoming the inflexion of the curve xS N r, and is located in the quadrants 1 and 3 respectively (see Fig. 3). The reason is as follows: d3r d3G = (-1 dxz +.=*@g ax3 +-y,Z.=*~fi a4G = 12p > 0 C--J ax4 p”=-E.p,z.=f;fl
(3
=o
Therefore, this inflexion is second-kind (stable) critical point, corresponding to the transition between two-step bistability and single stability. When p > 0, q < 0, p’ > -49, firstly keep p unchanged and q approaching to 0, in this case, the Midst stable branch will be gradually shortened. Under the limit conditions lim xy’ = fle &C q+o
= 0,
lim x:0) = liis(-@C) P-+0
= 0
the Midst branch is condensed to one point, that is the origin of the area of x, N r (x, = 0, r = 0). We have d2r a3G ax3 q=o,z.=o = (3dx;
(-1
this point becomes an inflexion, but becomes the connecting point of the The sub-second-kind critical point (r and the single bistability (or one-step
q=o, z.=o
b&able loops and completely singlestable curve (p = 3, q = -81/20) Fourth-kind
critical
a4G = -6p < 0 ( __ a24 1 q=o, 2.=0
belongs to sub-second-kind (unstable) critical point. It original upper and lower unstable branches (see Fig. 4). = 0, x, = 0) indicates the transition between tristability bistability).
Fig. 3 The transition between two-step
2.4
= 0,
Fig. 4 The transition between tristability and bistability (p = 3, q = 0)
points
When q 1 0, p2 > -49, there is always a single bistable curve as shown in Fig. 5 (when q > 0, the origin xS = 0, r = 0 are no longer critical points), the intercepts of High and Low branches on the xs axis are 6, -fl respectively. The threshold points on the left of the
42
Communications
in Nonlinear
Science & Numerical
Simulation
March 2000
High branch and the right of the low branch are zr) = &, xi-1 = -& respectively. The decrease of ] Q ] and p2 will shorten the area of the b&able curve, under the following limit conditions: lim XI+’ = lilie z$-’ = ;$ (+G) = ;lo wm = 0 (I’0 $-to
p2+o
p*-+0
p2-to
They are condensing to one point. At this time, the origin is back to the critical point, indicating the transition between the bistability and the single stability. As and the critical point K is fourth-kind (stable) critical point, which is the highest order of critical point in the tristability system.
(-;;)K
= 120 > 0
*.’
3 Remark K Based on the above classification of critig 0 cal points in the multistability chemical reaction system, we discussed the phase transition and the critical points in the tristability autocatalytic reaction under the non-ideal conditions, -2.0 and revealed the variety of the phase transition -25 0 and the critical points in the multistability chemr ical reaction system sufficiently. With the same method, we demonstrated that the higher the orFig. 5 The transition between two-step der of the multistability, the more complex and b&ability and single stability abundant the critical points are. cp = 0, q = 0) Both equilibrium phase transition and nonequilibrium phase transition are the break of some physical variables happening in the system, and are usually accompanied by the sequence change. But the mechanisms of phase transitions are different. The equilibrium phase transition comes from the reciprocity between the molecules; the non-equilibrium phase transition depends on the macro-kinetics of the non-equilibrium. The content of the non-equilibrium is much abroad, since the nonlinear non-equilibrium system has much more movement format than the equilibrium system. In this paper, the multistability chemical reaction phase transition and the classification of the critical points are the reflection of this phenomenon. Like the equilibrium phase transition, there lies some abnormal physics-chemical characteristics around the critical points of the non-equilibrium phase transition, and they are much more abundant. To understand and apply the abnormal characteristics around the variety of critical points will significantly improve the security and stability especially in chemical industry.
References [l] Vidal, C. and Pacauit, A., Nonlinear Phenomena in Chemical Dynamics, Springer-Verlag, Berlin, 1981 [2] Li Rusheng, Non-equilibrium Thermodynamics and Dissipative Structures, Tsinghua Univ. Press; Beijing, 1996 (in Chinese)
[3] Gao Qingyu, Cai Zunsheng and Zhao Xuezhuang, Nonlinear chemical reaction dynamics, Progress in Chemistry,
1997, 9(l):
59 (in Chinese)
Vol. 5, No. 1
GUO et al.: Blow up problem for Landau-Lifshitz
43
equations in two dimensions
[4] Li Rusheng, Equilibrium and Non-equilibrium Statistical Mechanics, Tsinghua Univ. Press, Beijing, 1995 (in Chinese) [5] Wu Fugen, Zhan Yehong and Wu Tingwan et al., The fluctuation of bistability in chemical reactions, Acta Physico-Chimica Sinica, 1998, 14(7): 659-663 [6] Zhan Yehong, Wu Fugen and Ou Fa, Critical phenomena of chemical reactions and its secondorder-like transition character, Chinese J. Chem. Phys., 1999, 12(l): 57-62 [7] Xu Zuyao, The Principle of Phase Transition, Science, 1988 (in Chinese) [8] Ou Fa and Wu Fugen, Critical phenomena of the bistability in chemical reactions and Landau theory of phase transitions, Acta Chimica Sinica, 1996, 54: 218 (in Chinese) [9] Wu Fugen, Zhan Yehong and Wu Tingwan et al., The slowing down in the bistability chemical reaction, J. Molecular Sci., 1998, 121(4): 199-203 [lo] Hu Gang, Stochastic Forces and Nonlinear Systems, Shanghai Scientific and Technological Education Press, Shanghai, 1994 (in Chinese) [ll] Xin Houwen, Reaction Kinetics in Fractal Media, Shanghai Scientific and Technological Education Press, Shanghai 1997 (in Chinese) [12] Reichl, L. E., A Modern Course in Statistical Physics, Univ. of Texas Press, Texas, 1980 [13] Landau, L. D. and Lifshitz, E. M., Statistical Physics, Pergamon, Oxford, 1958
Blow up problem for Landau-Lifshitz in two dimensions
equations
Boling GUO’, Yongqian HAN* and Ganshan YANG** *Institute of Applied Physics and Computational Mathematics, Nonlinear Center for Studies, P. 0. Box 8009, Beijing 100088, China ** Graduate School of China Academy of Engineering Physics, P. 0. Box 2101, Beijing 100088, (Received
March
China
7, 2000)
Abstract: The solutions of two dimensional Landau-Lifshitz equations, finite time, are obtained. Keywords: blow up, two dimensional Landau-Lifshitz equations
which blow up in
In 1986, Zhou and Guo in [l] proved the global existence of weak solution for generalized Landau-Lifshitz equations without Gilbert term in multi-dimensions. They consider the homogeneous boundary problem Z(x,t)
= 0,
for x E dR, 0 5 t 5 T
(1)
with the initial value condition
Z(x,O) = $4x), for the system of ferromagnetic
for x E !2
(2)
chain with several variables Zt = 2 x AZ + f(x,
t, Z)
(3)
where f(x, t, Z) is a given S-dimensional vector function in x E Rn, t E R+, Z E R3, p(x) is a given 3-dimensional initial value function on 0, R is a bounded domain in n-dimensional Euclidean space Rn.