Nonequilibrium phase transition to a time-dependent probability density for a model of charge-density waves

PHYSICAL REVIEW BVOLUME 35, NUMBER 71MARCH 1987 Nonequilibrium phase transition to atime-dependent probability density for a model of charge-density waves Luis L. Bonilla Departamento de Fi'sica Teorica, Universidad de Sevilla, Apartado Correos 1.065, Sector Sur, E-41080 Sevilla, Spain (Received 18 June 1986; revised manuscript received 10 September 1986) Anonequilibrium phase transition to astable time-periodic one-particle probability density is found in amodi6ed Fukuyama-Lee model of charge-density waves. The classical single-phase model of Gruner, Zawadowski, and Chaikin is derived in the zero-temperature limit as the equation for the order parameter of this transition. Two-time correlations are given as functionals of the order parameter. These results raise the question of the existence of stable nonequilibrium probability densities in other models. Stable time-dependent densities are shown to exist for replicas of general dynamical systems having astable time-dependent attractor in the zero-noise limit. This is so if the replicas are coupled via amean-6eld interaction and the thermodynamic limit is taken. I. INTRODUCTION Some time ago, Desai and Zwanzig studied critical dynamics of the mean-field version of the ptheory. They derived atime-dependent nonlinear Fokker-Planck equation for the one-particle probability density in the thermodynamic limit. Dawson proved this equation to be exact. The phase transition appeared as apitchfork bifurcation for the equilibrium probability density when the temperature (which is the square of the strength of the thermal noise) was varied. This result could never have been obtained in astandard linear Fokker-Planck equation, because the 0theorem implies the uniqueness and stability of the equilibrium. For amodification of the Fukuyama-Lee model of charge-density waves (CDW's), another nonlinear Fokker-Planck equation holds in the thermodynamic limit. This time, adifferent kind of phase transition is found in the zero-temperature limit: As the external field surpasses acritical value, the one-phase probability density changes from an equilibrium distribution to astable time-periodic (nonequilibrium) distribution. Furthermore, the order parameter of this nonequilibrium phase transition obeys the single-phase equation of Gruner, Zawadowski, and Chaikin. From the order parameter, oneand two-time correlations can be obtained. These correlations cannot be obtained from the original deterministic model of Ref. 4. The results for the simplified Fukuyama-Lee model raise the question of the existence of nonequilibrium probability densities and nonequilibrium phase transitions in general models. We give apartial answer: Let dx/ dt =f(x) be adynamical system having atime-dependent attractor x=A(t). Add avanishingly small white-noise forcing term to it. Consider Nreplicas of the resultant stochastic system coupled via amean-field interaction in the limit N~, the one-system probability density tends (as t~) to atime-dependent functional of A(t). II. THE SIMPLIFIED FUKUYAMA-LEE MODEL Adiscretized mean-field version of the Fukuyama-Lee model of CDW's consists of the following equations (j=l, ...,N): d&J./dt =E— hj sin(p~ p~ )— N — JP— N'gP +F' w. (t) . B,p(t,y) =, 'Fr), 'p(t, y)— — a,fR — hsing+ J(p(t) — y)]p(t,p)] (2) y(t) =pp(t, y)dy, p(0,y) =p(y), (4) +2Kp(t,y)dy=l .(5) Equation (2) has the form of aFokker-Planck equation, except that the coefficient p(t )is afunctional of the density. Once (2) is solved for an arbitrary function p(t), the consistency condition (3) acts as abifurcation equation, as we show below. The solution of (2) and (3) has the symmetry (3) p(t,y;y(t)) =p«,y+2rr;y(t)+2rr) . We use this symmetry as aboundary condition and solve Here QJ is the slowly varying phase of the CDW at the site j. Jis the stiffness of the CDW and Ethe applied electric field. hz and PJ represent the random impurity potential and pinning angle, respectively. The Gaussian white noise wj(t) [(wj) =0,(w;(t)wj(t')) =6's8(t — t')] contains the effect of the temperature. Equation (1) is due to Fisher. We want to consider the case hj -h &0and P~ 0in (1). The random external fields introduced by Fisher thus disappear. As we will show elsewhere, the resultant simplified model is still useful to analyze (1) in the strongpinning regime, where J«h. Suppose that the initial N-phase probability density corresponding to (1) (with h~ =h &0and PJ =0) is aproduct of None-phase probability densities p(pj), j=1,...,N. We have then proved that, asymptotica11y as N~, the one-phase probability density p(t,p) obeys the following nonlinear equations: 35 3637 1987 The American Physical Society 3638 BRIEF REPORTS 35 E=h sinpp .(9) (2) and (3) for phases in [0,2tr]. Equations (4) and (5) are initial and normalization conditions, respectively. Asymptotically for Fnear zero, the stationary solutions of (2) and (3) have the form 2x (~.~) — e2v(y;— ~)IF e2v(gy— )/Fd (6) ps ,~0 V(P;P) =J(P;P) EP — — hcosp, (7) f+ 2X (8) The integrals in (6) and (8) may be asymptotically evaluated by Laplace's method for Fnear zero. We have to find the minimum of V(p;(7() for pin [0,2tr]. For afixed p, 8(V(P;P) =0 has zero, one, two, or three different solutions in [0,2z], according to the relative values of the parameters p+E/J and h/J. Of these solutions, only one will, in general, minimize V(p;p). Let us call it pp. Then (7( — pp according to (8). Inserting this consistency condition in 8&V(p;p) =0, we find that For E&h, this equation has only one solution in [O,tr/2], which minimizes V(p;p). For E&h, (9) has no solution. Where then does an initial density evolve as s~'? To solve (2)-(5) in the low-temperature limit (F 0), we use the Wentzel-Kramers-Brillouin (WKB) method, thereby inserting in (2) the following function: p(t, y) =exp[ — e(t,(()/F] x[Zp(t, (()+FZ&(t,(()+O(F')] .(io) p(t) in (11)is given by (3) and (10). The solution of the initial-value problem for +is attained by the method of characteristics. It is We find an eikonal equation for Wand linear transport equations for the Z's. The eikonal equation is a,~+ —, '(a,+)'+[E— hsin(P — P)+J[(((t)— (t]]a,+=0 . ttr W(t,P;h,P) =Q(Z(g, t))+ —, '[0'(Z(t,g))] „dsexp 2Js+2h cos[@(Z(p,t),r) P]dr— (i2) e(s,0) =s .(i4) Once @(s,t) is known, Z(g, t) is its inverse function (if it exists): @(Z(((,t),t) =y, Z(@(s,t),t) =s .(is) To evaluate p(t) we use Laplace's method in (8). Asymptotically in the limit F0, p(t) is equal to the function p=pp(t )which minimizes +(t,p) Thus p.(t )satisfies Here Q(p) =+(O,p) — — Flnp(p), 0'(p):dQ(p)/d— p, and @(s,t) is the solution of the characteristic equations d@(s,t)/dt =E— hsin(@ — P) +J[p(t) — @]+0'(s) pt xexp Jt +hJcos [@(s,r)— P]dr, (13) 0 I tionary density p, (p;p) of (6). For E&h, p(t) is periodic and we can calculate an approximation to the density p(t,p) as follows. Suppose Eis larger than, but not too close to, h. Consider the following time-periodic Gaussian density, p(t,p)-exp[ —~(t — p(t) ~/[FW(t)] -O(~y-y(t) ~')l . This corresponds to having the exponent y(t) I'/[FW— (t)]+O(I(t — (((t) I') (i8) in the WKB solution (10). Let us insert this functional in the eikonal equation (11),and then expand sing=sing(t)+cos((7(t)[p — p(t)]+O(~ p— p(t) ~). 8,+(t,y(t)) =0 .(16) By ignoring 0(~p— p(t) ~)terms, we find the equation for W(t), Assuming that Z(p, t) exists, and considering (12), (16) is satisfied if 0'[Z(P(t),t)] =0. Suppose that cr is an extremum of the initial-value function A(s); think of an initial probability density peaked at p=o, for example. Then Z(p(t), t) =o, which gives N(cr, t) =p(t). By insertion in (13)and (14),we find dW(t )/dt +JW (t )=1+cosp(t ). This equation has the following T-periodic solution: tt W(t) =y(t)[l — y(T)] '„ds/y(s), (i9) (20) dP(t)/dt E— hsing(t), p(0) =o .(i7) y(t )=exp — 2Jt — 2h Jcosp(s )ds 0 Equation (17) is just the single-phase model of Griiner, Zawadowski, and Chaikin. For E~h, there are two stationary solutions of (17), (tp and tr — pp, where (tp is the value (9). pp is an asymptotically stable solution of (17) while tr — (tp is unstable. At E=h, pp=tr/2, and the two stationary solutions coalesce. For E&h, there are no stationary solutions of (17): For any o, (t7(t) is periodic, with period T=2tr(E — h)'t .Which p(t, p) do the solutions of (17)correspond to for different values of E/h? For E~h, p(t) tends to pp, and p(t, p) tends to the staf+ 2X f+ 2X &y;(t)yj(t')) =b;,„„y(harp'(t — t';y, y)dydee .(2i) Equations (18)and (20) form anonuniform approximation of p(t, p) which is asymptotically valid for pnear p(t ) and F0. W(t) is approximately equal to the correlation (p (t)). Near E=h, W(t) diverges. We need to keep higher-order terms in (18), typically up to third-order terms. Two-time correlations (p(t )p(t ')) are calculated by means of 35 BRIEF REPORTS 3639 p'(0;y, y) -b(y — y) . Laplace's method applied to (21) yields (22) (yr(r)y, (r')&-br, gtily(r;y)dy . p(r;y) obeys Eq. (17) with initial condition p(t;y) =y. (23) III. AGENERAL RESULT FOR MEAN-FIELD MODELS IS IN THE WEAK-NOISE LIMIT Let us prove here the claim made in the Introduction concerning dynamical systems dx/dt =f(x), with an attractor A(t). We now add awhite noise term to f(x), and analyze the corresponding equation (1)with f(xj), substituting E— hj sin(&J — pj). The analysis of Sec II . goes through with trivial modifications. It yields the equation dx(t)/dt =f(x(t)), x(0) =a, for the mean value of x. As 4(t) is the asymptotica[iy stable solution of this equation, x(t) A(t) as t~. The probability density will tend to atime-dependent functional of A(t) which is of the form (18) for xclose to A(t). It seems reasonable to ask that the local correlation W(t) have the same features as A(r). Further work in this direction [for an aperiodic A(r ), for example, astrange attractor] seems interesting. IV. DISCUSSION We have analyzed asimplified Fukuyama-Lee model of CDW's in the low-temperature limit. This model retains the mean-field interaction among the phase of the CDW at different sites, but it ignores the external noise due to the impurities. In the thermodynamic limit, we obtain a nonlinear equation for asingle-phase probability density. An asymptotic analysis valid for F0(vanishing temperature), yields the classical equation (17), for the mean phase of the CDW. Thus the single-phase model of Ref. 4can be derived from amodel still containing collective effects due to the interaction among phases at different sites. Had we included the effect of the noises, we would have obtained Fisher's equation (4.4),5with adifferent meanThe conditional density p'(t;p, y) satisfies Eqs. (2), (3), and (5), with the initial condition (22) below instead of (4): ing of the variables: The phase pis now acollective coordinate which minimizes the WKB exponent +, not any single phase pJ as in Ref. 5. This difference turns out to be crucial for the analysis. Assuming (as we do here) that the phases of the deterministic pinning potential are coherent (pj =O,hj =h), our analysis shows the following. (a) To the extent that the classical model of Ref. 4is reasonable, 'the depinning of the CDW is anonequilibrium phase transition; the probability density changes from astationary equilibrium distribution to atime-periodic one. (b) Our statistical analysis yields expressions for the mean and different correlations of the phase of the CDW. All of them are periodic with the same period as the mean. (c) The critical exponents for our model are, of course, those calculated in Ref. 4. They are incompatible with experiments. To get better results we must study Eq. (1)including the noises hj and pj. Furthermore, we have shown that nonequilibrium timedependent probability densities are conspicuous in meanfield models: In the weak-noise limit, we can build timedependent densities out of any dynamical system having a time-dependent attractor. The main requirements for this are the mean-field interaction among replicas of the dynamical system, the molecular chaos initial condition for the ¹eplica probability density, and the thermodynamic limit. If extrapolation from this result is legitimate, nonequilibrium phase transitions present in the weak-noise limit should also be present for finite temperature. We have checked this statement in amodel of synchronization of oscillators, where standard Hopf bifurcation techniques work for any noise strength. " Note added in proof. When V(p;p) has no minimum for afixed p, there is asolution of the form (6), which has asingle maximum at /=2'. This stationary solution, which is overlooked here, is crucial to explaining why there are no sliding CDW's at high temperatures. ACKNOWLEDGMENTS The author is indebted to Dr. W. Wolff for introducing him to CDW models and for collaboration in related storks, to Professor J. B. Keller for helpful comments, and to Professor M. Morillo for valuable discussions and for bringing Refs. 1and 2to his attention. ~R. Desai and R. Zwanzig, J.Stat. Phys. 19, 1(1978). 2D. A. Dawson, J.Stat. Phys. 31,29 (1983). 3H. Fukuyama and P. A. Lee, Phys. Rev. B17, 535 (1977). G. Griiner, A. Zawadowski, and P. M. Chaikin, Phys. Rev. Lett. Lett. 46. 511 (1981). SD. S. Fisher, Phys. Rev. B31, 1396 (1985). sL. L. Bonilla (unpublished). 7L. L. Bonilla, J.Stat. Phys. (to be published). sThe difference between (6) and the true 2~-periodic density is exponertially small as F0, namely, O(exp[ — 4+1 x(a+p)/F]) physically, the probability current between each period [which is needed to guarantee the periodicity of p, (p;p)l is exponentially small, and it can safely be ignored, as we did in (6). This is due to the symmetry of the saddle-node bifurcation, in which two stationary states coalesce and disappear. See M. Mangel, SIAM J. Appl. Math. 36, 544 (1979), where the corresponding (linear) Fokker-Planck equation is analyzed. In our case we must supplement Mangel's analysis with the consistency condition (3). 'oG. Griiner and A. Zettl, Phys. Rep. 119, 117 (1985). "L.L. Bonilla, J. M. Casado, and M. Morillo (unpublished).