Holes in the probability density of strongly colored noise driven systems |
| |
Authors: | G Debnath Frank Moss F Marchesoni Th Leiber H Risken |
| |
Institution: | (1) Department of Physics, University of Missouri at St. Louis, 63121 St. Louis, Missouri;(2) Dipartimento di Fisica, Universita' degli Studi e CISM, I-06100 Perugia, Italy;(3) Theoretische Physik, Universität Ulm, D-7900 Ulm, Federal Republic of Germany |
| |
Abstract: | A qualitative change in the topology of the joint probability densityP( ,x), which occurs for strongly colored noise in multistable systems, has recently been observed first by analog simulation (F. Moss and F. Marchesoni,Phys. Lett. A
131:322 (1988)) and confirmed by matrix continued fraction methods (Th. Leiber and H. Riskin, unpublished), and by analytic theory (P. Hänggi, P. Jung, and F. Marchesoni,J. Stat. Phys., this issue). Systems studied were of the classx=– U(x)/ x+ (t, ), whereU(x) is a multistable potential and (t, ) is a colored, Gaussian noise of intensityD, for which ![lang](/content/k50p224057761k37/xxlarge9001.gif) ![epsi](/content/k50p224057761k37/xxlarge949.gif) =0, and ![lang](/content/k50p224057761k37/xxlarge9001.gif) (t) (s) =(D/ )exp(– t–s / ). When the noise correlation time is smaller than some critical value 0, which depends onD, the two-dimensional densityP( ,x) has the usual topology P. Jung and H. Risken,Z. Phys. B
61:367 (1985); F. Moss and P. V. E. McClintock,Z. Phys. B
61:381 (1985)]: a pair of local maxima ofP( ,x), which correspond to a pair of adjacent local minima ofU(x), are connected by a single saddle point which lies on thex axis. When > 0, however,the single saddle disappears and is replaced by a pair of off-axis saddles. A depression, or hole, which is bounded by the saddles and the local maxima thus appears. The most probable trajectory connecting the two potential wells therefore does not pass through the origin for > 0, but instead must detour around the local barrier. This observation implies that successful mean-first-passage-time theories of strongly colored noise driven systems must necessarily be two dimensional (Hänggiet al.). We have observed these holes for several potentialsU(x): (1)a soft, bistable potential by analog simulation (Moss and Marchesoni); (2) a periodic potential Th. Leiber, F. Marchesoni, and H. Risken,Phys. Rev. Lett.
59:1381 (1987)] by matrix continued fractions; (3) the usual hard, bistable potential,U(x)=–ax
2/2+bx
4/4, by analog simulations only; and (4) a random potential for which the forcingf(x)=– U(x)/ x is an approximate Gaussian with nonzero correlation length, i.e., colored spatiotemporal noise, by analog simulation. There is a critical curve 0(D) in the versusD plane which divides the two topological behaviors. For a fixed value ofD, this curve is shifted toward larger values of 0 for progressively weaker barriers between the wells. Therefore, strong barriers favor the observation of this topological transformation at smaller values of . Recently, an analytic expression for the critical curve, valid asymptotically in the small-D limit, has been obtained (Hänggiet al.).This paper will appear in a forthcoming issue of theJournal of Statistical Physics. |
| |
Keywords: | Topology of probability density colored noise noise-induced topologies analog simulation matrix continued fractions bistable potential bistability random potential noise-correlation-time-induced transition critical transition spatiotemporal noise |
本文献已被 SpringerLink 等数据库收录! |
|