PHYSICAL REVIEW A VOLUME 41, NUMBER 4 15 FEBRUARY 1990 Theory for relaxation at a subcritical pitchfork bifurcation P. Colet, F. De Pasquale, * M. O. Caceres, and M. San Miguel Departament de I'I'sica, Universitat de les Ilies Balears, E-07071, Palma de Mallorca, Spain (Received 18 August 1989) A theory is presented for the transient process of decay from the state of marginal stability occur- ring in a subcritical pitchfork bifurcation. We introduce an approximation for the stochastic paths that allows the calculation of a generating function for the statistics of the escape time. The time- dependent moments and transient fluctuations are calculated from the distribution of passage times by a scaling transformation that is associated with initial non-Gaussian statistics. Our results are in agreement with numerical simulations reported also here. I. INTRODUCTION Nonlinear systems away from equilibrium exhibit a variety of instabilities when the appropriate control pa- rameters are changed. By such changes of control pa- rameters the system can be placed in a stationary state which is not globally stable. The system, in general, will then relax to a global, stable state. This transient process of relaxation is triggered by fluctuations. The statistical description of such a transient process is one of the main subjects in the study of nonequilibrium phenomena where noise plays a crucial role. Two complementary descrip- tions of transient processes exist. In one of them the focus is on the calculation of the lifetime of the initial state: The time at which the system leaves the initial state is a random quantity whose statistics can be general- 1y calculated by first-passage-time techniques. ' A second description focuses on the time-dependent averages of the relevant variable (order parameter) which is used to de- scribe the instability. Fluctuations measured as a time- dependent variance during the transient process are known to be orders of magnitude larger than at the initial and final states. Generally speaking we can distinguish three regimes in the decay process from a state which is deterministically stationary but decays due to fluctuations. In the first re- gime the system is close to the initial state. In the second the system leaves that state, and in the third the final steady state is approached. Noise effects are obviously important in the first regime, while in the second they have a relatively small influence on the evolution of the system which is essentially deterministic. Fluctuations appear again around the final steady state, but they are often quite small. A detailed description of a relaxation process depends on the nature of the initial state and the type of instability involved. A well-studied case ' is the one of a supercritical pitchfork bifurcation. In this situ- ation an initially stable state becomes unstable when the instability is crossed. The initial stage of the relaxation process is associated with linear stochastic dynamics, so that an initial description in terms of Gaussian statistics is possible. The time scale separating the first and second regimes is the lifetime of the state calculated as a mean first-passage time (MFPT). The time regime connecting the linear and nonlinear stages of evolution exhibits dynamical scaling. Scaling implies that a good descrip- tion of the first and second regimes is obtained by the solution of the deterministic evolution equation with ini- tial conditions which are assumed to be random variables with a Gaussian distribution. The relaxation process is much less well understood when the initial state is one of marginal stability. These states appear in first-order-like instabilities at the end point of hysteresis cycles. Typical cases are those of a saddle-node instability and a subcrit- ical pitchfork bifurcation. The essential difficulty of the description of a relaxation from a state of marginal stabil- ity is that there is no regime of interest in which a linear approximation is meaningful. The process requires a nonlinear stochastic description right from the beginning. As a consequence, no initial Gaussian regime exists. A manifestation of this fact is the phenomenon of transient bimodality discussed ' in chemical and optical systems and observed ' in optical and electronic systems: In the supercritical pitchfork bifurcation an initial Gaussian dis- tribution broadens and eventually becomes double peaked during the nonlinear regime. ' In the relaxation from a marginal point, the probability distribution continues to have a well-defined peak centered at the initial state while other peaks appear during the transient. There are few theoretical studies of relaxation from states of marginal stability. ' They are mostly devoted to a direct calculation of passage-time statistics for the saddle-node instability. An interesting contribution go- ing beyond a passage-time calculation is that of Ref. 14, where a connection between the transient time-dependent variance and the passage-time distribution is established. A serious drawback of the present theoretical under- standing is the lack of a direct approximation for the sto- chastic paths occurring in the decay process. Such an ap- proximation is available ' for the relaxation of an unsta- ble state in the supercritical pitchfork bifurcation. This allows a precise understanding of the decay process. In the case of an initial marginal state an additional difficulty occurs in the saddle-node bifurcation besides the absence of a linear regime for the escape process. In the subcritical pitchfork bifurcation, and close to the in- 41 1901 1990 The American Physical Society