The quasideterministic approach in the decay of rotating unstable systems driven by gaussian colored noise

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1 REVISTA MEXICANA DE FÍSICA 48 SUPLEMENTO SEPTIEMBRE The quasideterministic approach in the decay of rotating unstable systems driven by gaussian colored noise J.I. Jiménez-Aquino and M. Romero-Bastida Departamento de Física Universidad Autónoma Metropolitana-Iztapalapa Apdo. Post México D.F. Mexico Recibido el 19 de marzo de 1; aceptado el 1 de mayo de 1 The quasideterministic (QD) approach and passage time (PT) distribution are proposed to characterize the decay process of rotating unstable systems submitted to the action of gaussian colored noise and constant external force. The Langevin type equation is formulated in two spaces of coordinates represented by the x and y vectors being y a transformed space of coordinates obtained through a time dependent rotation matrix which leaves the norm invariant. We study the systems of two variables and show that the validity limit of QD approach is that for which the amplitude of the external force is less or the same order than the intensity of internal noise. In this limiting case the numerical simulation shows that the rotational effects of those systems are practically neglected. The theoretical result of the mean passage time is compared with those numerical results for small correlation times. Keywords: inestabilities; colored noise; rotation matrix; Langevin equation; fluctuation Se proponen los tiempos de paso (PT) y la aproximación quasideterminista (QD) para caracterizar el proceso de decaimiento de sistemas inestables rotantes sometidos a la acción de fuerza externa constante y ruido de color gaussiano. Se formula la ecuación de Langevin en dos espacios de coordenadas representados por los vectores x y y siendo y un espacio transformado de coordenadas que se obtiene a través de una matriz de rotación dependiente del tiempo el cual deja a la norma invariante. Estudiamos los sistemas de dos variables y mostramos que el límite de validez de QD es aquel en el que la amplitud de la fuerza externa es menor o igual que la intensidad del ruido interno. En este caso límite la simulación numérica muestra que los efectos rotacionales de esos sistemas son prácticamente despreciables. Los resultados teóricos de los tiempos de paso son comparados con esos resultados numéricos para tiempos de correlación pequeños. Descriptores: inestabilidades; ruido coloreado; matriz de rotaciones; ecuación de Langevin; fluctuaciones PACS: 5.4.j 1. Introduction The transient behaviour of non-equilibrium systems [1 ] particularly the decay of unstable states in which fluctuations play an important role [ ] still offers new perspectives. This is because very recently in Refs. 1 and 11 it has been proposed a new mathematical scheme to characterize through the passage time (PT) distribution the decay process of rotating unstable systems submitted to the influence of both internal fluctuations (internal noise) and constant external force. These are systems which once leaving the initial unstable state by effect of internal fluctuations practically describe deterministic and rotating trajectories. Such is the case of the laser system as that studied in Refs. 6 9 which is a particular case of such rotating unstable systems; although in that references the rotating character of the laser system was not exhibited explicitly. According to Refs. 1 and 11 the matricial formalism has been developed in the context of two dynamical representations for the Langevin type equation and applied to the study of the decay process of rotating unstable systems of two variables. One dynamical representation is described in x space of coordinates and the other in a transformed y space of coordinates. The latter is obtained by means of a timedependent rotation matrix which leaves the norm invariant. Two limiting cases corresponding to weak and large amplitud of the external force compared with the intensity of internal noise have been studied in those references. The case of weak amplitude is studied in Ref. [1] and corresponds to that situation for which the amplitude of the external force is less or equal than the intensity of internal noise. In this case it is shown in y space of coordinates by numerical simulation that the dynamical trajectories described by the system to reach a reference value are approximately straight lines. In other words if the intensity of the internal noise dominates over that of the external force then the rotating effects of the system are practically neglected and therefore the dynamical evolution followed by the system is nearly described by straight lines. It is precisely in this limiting case where QD approach is valid and is in the y scheme where it can be understood why this approach works well in the characterization of rotating unstable systems. The opposite case studied in Ref. 11 corresponds to large amplitude of external force and occurs when this amplitude dominates over the intensity of internal noise. In this case it is also shown by numerical simulation that the rotational effects of the system must be taken into account and therefore QD approach is no longer valid to describe the system. Our purpose in this paper is to characterize through the PT distribution the decay process of rotating unstable systems of two variables when the decay process is driven by both correlated exponentially internal noise (gaussian colored noise) and constant external force. We only study the case of weak amplitude of the external force and therefore the QD approach must be the appropriate scheme. Our proposal will

2 THE QUASIDETERMINISTIC APPROACH IN THE DECAY OF ROTATING be given in both x and y dynamical representations in order to verify the equivalence between both schemes taking into account that the y space of variables is the appropriate to understand the physics behind the QD approach. We will show in x scheme that the dynamical trajectory described by the system is a spiral on the plane (x 1 x ) whereas in the y scheme it describes loops around an imaginary axis on the plane (y 1 y ). On the other hand the dynamical relaxation of initial conditions driven by gaussian colored noise of non-rotating unstable systems is a problem already studied in Refs Because of non-markovian character of the problem the coupling between the initial state of the sysem at time t = and the noise appears in a natural way and therefore the statistical properties of the quantity x()ξ(t) are in general different from zero being x() the initial condition of some physical variable x and ξ(t) the noise. A similar situation occurs in the study of the dynamical relaxation of matricial systems of two and three variables as those proposed in Ref. 15 where the dynamical characterization has been given in terms of the nonlinear relaxation times and in the x dynamical representation. The study has been made from a theoretical point of view and no physical meaning was properly discussed. In this work we pay attention to this point. Because of mathematical simplicity we will only consider the case of statistical independence between the noise and the initial state of the system which will be taken as x() = and therefore x()ξ(t) =. In this sense the initial condition is not statistically distributed but fixed at the initial unstable state. We show that in the variance of effective initial conditions there is a coupling term between the rotating parameter and the correlation time to the second order in the latter. However if we take the first order of approximation in the correlation time this coupling effect disappears and then we recover the results for non-rotating unstable systems as reported in Refs Our theoretical predictions are compared with numerical simulation results.. The theoretical scheme.1. Dynamical representation in x space In space of x coordinates represented by n-physical variables the Langevin type equation which governs the dynamical behaviour of rotating unstable systems in the presence of constant external force can be written as ẋ = Ax N(r)x f e z(t) (1) where N(r) is a scalar function which accounts for nonlinearities due to the fact that r x = xx (the square modulus of vector x); the column vector f e is the external force represented by constant elements f ei ; z(t) is the fluctuating force with elements ξ i (t) such that ξ i (t) = and correlation function ξ i (t)ξ j (t ) = ɛ ij τ δ ije t t τ i j = 1... n () where ɛ ij is the corresponding noise intensity τ the correlation time and A is a n n matrix which satisfies the following: A = D W where D is a diagonal matrix D = ai with a > I is the unit matrix and W is a real antisymmetrix matrix. In this case the linear systematic force F = Ax of the dynamics [Eq. (1)] can also be written as F = F c F nc where F c is the conservative part Dx whereas F nc will be the corresponding nonconservative part Wx and therefore the total force F is not in general derived from a potential because F = Wx. So the rotating character of dynamics (1) appears in the properties of matrix W... Dynamical representation in y space The coupling between x variables of Eq. (1) can be removed if we make the change of variable y = e Wt x which is a time dependent rotation represented by the matrix R(t)=e Wt. So in the transformed space the Langevin equation can be written as ẏ = Dy N(r)y R(t)[f e z(t)] (3) where the scalar function N(r) is the same as before because the square modulus also satisfies r y = ỹy; that is it is an invariant under the transformation. As a consecuence of the transformation we can also see that the nonconservative part of the linear systematic force has been removed and rotating effecs of matrix W are actually coupled to the presence of both external (constant) and stochastic force. It is clear that the dynamics given by (3) represents a set of decoupled equations for each component y i because the matrix D is diagonal. In the next section it will be seen that R(t) satisfies the properties of an orthogonal rotation matrix and the statistical properties of the passage time will be the same in both dynamical representation due to the norm invariance..3. The QD approach and passage time distribution.3.1. Space of x variables In this representation the QD approach starts with the linear Langevin dynamics of Eq. (1) that is ẋ = Ax f e z(t). (4) The formal solution of this equation for zero initial condition is then where x(t) = e At h(t) = e at e Wt h(t) (5) h(t) = t e as e Ws [f e z(s)] ds (6) and therefore is a gaussian stochastic process. Rev. Mex. Fís. 48 S1 ()

3 164 J.I. JIMÉNEZ-AQUINO AND M. ROMERO-BASTIDA The QD approach tells us that h(t) plays the role of an effective initial condition in the limit of long times. This can be achieved by evaluating the integral in Eq. (6) which requires of the explicit expression of matrix W(t) and therefore of its diagonalization. Then for small values of each element of matrix z(t) we can guarantee that dh(t) lim = lim e at e Wt [f t dt e z(t)] (7) t and therefore h( ) becomes a constant matrix h whose elements h i are gaussian random variables. In this case the process (5) becomes a quasideterministic process which in terms of the r variable reads as r(t) = h e at (8) where h = hh is the square modulus of vector h(t) and plays the role of an effective initial condition. The mean passage time t required by the system to reach a reference value R is then t = 1 ( ) R a ln h. (9) This time scale can be calculated through the statistical properties of the random variable h. The statistics of random variable h must be obtained from the marginal probability density P (h) with the knowledge of the joint probabiltiy density P (h 1... h n ) = [ exp 1 1 (π) n/ (Detσ ij ) 1/ ] (σ 1 ) ij (h i h i )(h j h j ) (1) ij where the variance (i = j) and covariance (i j) of the matrix σ ij are defined as σ ij = h i h j h i h j. (11) The set of random variables h i are independent if the matrix σ ij is diagonal with elements σ ii = σ. Using the jacobian transformation dv = J(u) du with u = (u 1... u n ) a new space of variables the joint probability density in the space (h u... u n ) will be written in a formal way as P (h u... u n )dv = C exp[ α (h q qh)] dv (1) where u 1 = h and C is a constant. The parameters α and q are given by α = 1/σ and q qq = h 1 h n. Finally P (h) can be calculated if we know the jacobian and integrate over the rest of variables (u... u n )..3.. Space of y variables In this space of coordinates the QD approach starts also with the linear approximation of Eq. (3) ẏ = Dy R(t)[f e z(t)] (13) whose formal solution for zero initial condition is now where h(t) = t y(t) = e at h(t) (14) e as R(s)[f e z(s)] ds. (15) As in the x scheme the process h(t) plays the role of an effective initial condition in the limit of long times. To evaluate the integral (15) the explicit expression of the rotation matrix R(t) is required. So for small values of each element of matrix z(t) we can guarantee that dh(t) lim = lim e at R(t)[f t dt e z(t)] (16) t and therefore h( ) becomes a constant matrix h with elements h i. Again the process (14) becomes a quasideterministic process which in terms of the r variable reads r(t) = h e at (17) where h = hh is the square modulus of vector h(t). We can observe that Eqs. (15) and (6) are exactly the same. This is because the norm is invariant under the rotation R(t). The mean passage time required by the system to reach a reference value R is also t = 1 ( ) R a ln h. (18) The passage time (18) obtained in the y scheme is the same as that given by Eq. (9) obtained in the x scheme because the statistics of the variable h is the same in both dynamical representations due to Eq. (6) is exactly the same as Eq. (15). 3. The systems of two variables In the following we will consider that the initial condition is fixed on the initial unstable state this means that the initial state of the system is statistically independent of the noise and therefore x()ξ(t) = for x representation or y()ξ(t) = for y representation The dynamics in x space For this type of systems the Langevin equation (4) is such that A = D W and ( ) ( ) a ω D = W = (19) a ω where ω is the rotating parameter. Clearly the F = ωê 3 with ê 3 the unitary vector along Rev. Mex. Fís. 48 S1 ()

4 165 THE QUASIDETERMINISTIC APPROACH IN THE DECAY OF ROTATING... and therefore (h1 h ) are independent random variables. The marginal probabity density is then P (h) = α hi (α hq)e α (h q ) (3) where I (x) is the modified Bessel function of zeroth order. The time scale (1) associated with the linear dynamics (4) in the case of two variables is given by hti = hti 1 X ( 1)m m (β ) a m=1 mm! O(²) O(q ) O(²q ) (4) where β = α q with α = 1/σ σ is given by Eq. () and hti = F IGURE 1. Dynamical evolution of one trayectory of the system of two variables in the (x1 x )-space for values fe = 1. a = 3. ω = 1. R = 1. and ² = 1 4. axes x3. In Fig. 1 we show an example of the dynamical behaviour of the system in (x1 x ) space of coordinates. According to Eq. (6) the mean values of marix h read as hh1 i = hh i = afe1 a ω ωfe1 a ω ωfe a ω afe a ω hti = ht(τ = )i () (1) where fe = fe1 fe is the square modulus of vector fe. With the diagonalization of the matrix W and assuming that ²11 = ² = ² it can be shown that if ωτ h(1 aτ ) the covariance i 6= j and variance i = j of matrix σij are such that σ1 = σ1 = and σ11 = σ = σ where ² (ωτ ) σ = 1. () a(1 aτ ) (1 aτ ) (ωτ ) This expression contains an additional contribution to that obtained when there is no rotations i.e. if ω = we get the same result as that obtained in the study of non-rotating unstable systems. For small τ and ω 6= we get the same result for non-rotating systems because the additional contribution disappears. So that the matrix σij of Eq. (11) is diagonal with elements σ1 = σ1 = and σ11 = σ = σ with σ = σ τ O(τ ) (6) where ht(τ = )i is the corresponding passage time in the limit of white noise [1] that is 1 {ln(α R ) γ} a The parameter q q q = hh1 i hh i in this case will be given by f q = e a ω (5) is the passage time associated with Eq. (4) in the absence of external force and γ is the Euler constant. It can be show that at first order in τ the passage time reduces to ht(τ = )i =. 1 {ln(α R ) γ}. a 1 X ( 1)m m (β ). (7) a m=1 mm! In this case α = 1/σ such that σ = ²/a and β = α q. The dynamical characterization of Eq. (4) through Eq. (7) must be valid if the parameter β 1 which means that the amplitude of the external force must be less or the same order than the intensity of internal noise The dynamics in y space According to Eq. (14) we can verify that D= µ a a µ R(t) = cos ωt sin ωt sin ωt cos ωt (8) where R(t) is obtained from the diagonalization of matrix W and satisfies the properties of a rotation matrix; that e is R(t) = R 1 (t). This matrix also satisfies the properties of an orthogonal matrix. In this representation obviously F = Dy =. In Fig. we show an example of the Rev. Mex. Fı s. 48 S1 ()

5 166 J.I. JIME NEZ-AQUINO AND M. ROMERO-BASTIDA F IGURE. Dynamical evolution of one trayectory of the system of two variables in the (y1 y )-space for values fe = 1. a = 3. ω = 1. R = 1. and ² = 1 4. F IGURE 3. Dynamical evolution of one trayectory of the system of two variables in the (y1 y )-space for the same values as Fig. except that fe = ² = 1 4. dynamical trajectory in the (y1 y ) space of coordinates. Here it is appropriate to remark that this dynamical trajectory corresponds to the case in which the amplitude of the external force is larger than the intensity of the internal noise. From Eq. (16) we also shown that the mean values of matrix h(t) are afe1 hh1 i = a hh i = a ω ω ωfe1 ωfe a ω afe a ω (9) which are the same as those of Eq. () and therefore the parameter q is the same as (1). In a similar way we show that the matrix σij of Eq. (11) is diagonal with elements σ1 = σ1 = and σ11 = σ = σ where ² (ωτ ) σ = 1 a(1 aτ ) (1 aτ ) (ωτ ) (3) and therefore σij is also diagonal with elements σ1 = σ1 = and σ11 = σ = σ being σ = σ. The time scale to characterize the decay process of the dynamics (14) in the case of two variables will be the same as those given in Eqs. (6) and (7). The dynamical trajectory in the case of weak amplitude of the external force is shown in Fig. 3 in the (y1 y ) space of coordinates. As we can see in this limit of approximation the dynamical evolution of the system is practically a straigth F IGURE 4. Comparison between time scale [Eq. (6)] rescaled with the variable ln(1/²) aτ and numerical simulation for values fe = ² a = 3. ω = 1. R = 1. and different values of τ. The simulation results correspond to values of ² between 1 and 1 5. The straight line corresponds to the theoretical results Eq. (6); ( ) are the simulation results for white noise (τ = ); ( ) are simulation results for τ =.1; ( ) are for τ =. and (N) are for τ =.3. line; therefore the time scale (6) must be the appropriate the quantity to describe such a dynamics. This is corroborated by the results of Fig. 4 where we compare the theoretical prediction (6) with simulation results for different values of the involved parameters. In that Figure we show the time scale hti versus the scaling variable ln(1/²) aτ with excellent agree- Rev. Mex. Fı s. 48 S1 ()

6 THE QUASIDETERMINISTIC APPROACH IN THE DECAY OF ROTATING ment between the theoretical predictions and the computer simulation results. All of our data were obtained with the algorithm of Ref. 16 originally designed to deal with one-dimensional systems driven by multiplicative white and colored noise. We considered the specific case of additive noise and extended the algorithm to the case of two variables. It was necessary to consider contributions of second order in the integration step in contrast with the case of white noise reported in Ref. [1] in which the first order contributions were enough in the simulation results. 4. Concluding remarks The QD approach in colored noise problem to characterize the decay process of rotating unstable systems of two variables is also a good approximation in the limiting cases of weak amplitud of external force and small correlation time. The problem has been studied in the x and y dynamical representations showing that the mean passage time is the same in both dynamics due to the norm invariance. The above characterization is better understood in the y dynamical representation according to simulation results shown in Fig. 3. The expression of the variance () contains a coupling term between the rotating parameter ω and the correlation time τ which contributes to the second order in the correlation time. This coupling term is neglected if we take the first order of approximation in τ. In this case the variance will be the same as that studied in Refs for non-rotating unstable systems with colored noise. This is an expected result because QD approach is only valid in the limit of weak amplitude of external force in which the rotational effects are neglected. The theoretical expression (6) has been rescaled to the white noise limiting case and compared with simulation results showing an excellent agreement. It would be interesting to study the problem with colored noise but in the case in which the amplitude of the external force dominates over the intensity of internal noise and compare with that studied in Ref. 11. In this case the rotational effects must be taken into account and therefore another approach must be required. Acknowledgments Financial support from Consejo Nacional de Ciencia y Tecnología (CONACyT) México is acknowledged. 1. C. Vidal and A. Pacault Nonequilibrium Dynamics in Chemical Systems (Springer Verlag 1984).. H.L. Swinney and J.P. Gollub Hydrodynamic Instabilities and the Transition to Turbulence (Springer Verlag 1981). 3. F.T. Arecchi V. Degiorgio and B. Querzola Phys. Rev. A 3 (1971) M. Suzuki Phys. Lett. A 67 (1978) M.C. Torrent and M. San Miguel Phys. Rev. A 38 (1988) S. Balle F. de Pasquale and M. San Miguel Phys. Rev. A 41 (199) G. Vemuri and R. Roy Phys. Rev A 39 (1989) J. Dellunde M.C Torrent and J.M. Sancho Opt. Comm. 1 (1993) J. Dellunde J.M. Sancho and M. San Miguel Opt. Comm. 39 (1994) J.I. Jiménez-Aquino and M. Romero-Bastida Physica A (1) to be published. 11. J.I. Jiménez-Aquino Emilio Cortés and N. Aquino Physica A (1) to be published. 1. J.M. Sancho and M. San Miguel Phys. Rev. A 39 (1989) J.I. Jiménez-Aquino J. Phys. A 7 (1994) J. Casademunt J.I. Jiménez-Aquino and J.M. Sancho Phys. Rev. A 4 (1989) J.I. Jiménez-Aquino Physica A 45 (1997) J.M. Sancho M. San Miguel S.L. Katz and J.D. Gunton Phys. Rev. A 6 (198) Rev. Mex. Fís. 48 S1 ()