VOLUME 74, NUMBER 22 Fluctuations and Fracture M. Marder Department of Physics and Center for Nonlinear Dynamics, The University of Texas at Austin, Austin, Texas 78712 (Received 4 November 1994; revised manuscript received 4 April 1995) This Letter describes the way that metastable systems escape from traps. Brief periods during which the system apparently runs backwards in time play a crucial role. These ideas are applied to a model for one of nature's most obviously irreversible phenomena, the shattering of a body by a crack. PACS numbers: 82.20.Db, 46.30. Nz, 62.20.Mk vi + dxi ~ vi — F; + bmv; + kTb g, vi (2) where b is a phenomenological parameter describing the rate at which thermal equilibrium establishes itself, k is Boltzmann's constant, and T is the temperature. Equation (2) is of a form whose solution may be rep- resented by a functional integral [7]. A brief calculation shows that the probability of a state beginning at x, v and ending at xf, vf after time 7. is g(xf, vf: 7) = — J dt g [mx; — F;+mbx;] /4mkTb 0 In Eq. (3) the integral is over all paths having the property that x, (0) = x, and x;(0) = v, , f ~ f x;(r) = x; and x, (T) = v, . (5) In December of 1947, the 5. 5. Ponagansett suddenly cracked in half while floating quietly at its pier in Boston [1, 2]. It provided dramatic evidence that macroscopic objects can irreversibly be changed by imperceptible fIuctuations. The purpose of this Letter is to present a general way to calculate the likelihood of such a fIuctuation, and to illustrate the formalism in a simple model of fracture. I will show that the formal study of Hamiltonian sys- tems in thermal baths leads one to search for trajecto- ries in phase space which move from specified initial to specified final conditions while minimizing deviation from Newton's laws. When a metastable system escapes from a trap, initial stages of the process develop as if the sys- tem were moving backwards in time [3]. General formalism — Consid. er a collection of I par- ticles of mass I at positions x;, moving in response to forces F;(xi, . .. , xM), with velocities v;, and which are characterized statistically by g(X1, Vi, X2, V2, .. . , XM, VM ' . t) . The distribution function g gives the probability that particles i are at positions x; with velocities v; at time t. When placed in contact with a heat bath, this probability distribution is assumed to obey the Fokker- Planck equation [4 — 6] U(T) = min x(t) 4bX (7) with the deviation D; from Newtonian mechanics D;(t) = — mx; — [F; — mbx;]. The minimization in Eq. (7) is carried out over all paths x(t) starting and ending with the positions and velocities required by Eqs. (4) and (5). Two types of minima of Eq. (7) can be understood analytically. The first is a path x; ' '(t) which obeys Newton's laws, so that D;(t) = 0. The activation barrier U associated with this sort of path is zero, but it cannot in general satisfy both boundary conditions Eqs. (4) and (5). In particular, such paths cannot solve. problems in which systems escape from local energy minima, since the damping in Eq. (8) requires energy to decrease as t moves from 0 to 7. A second class of minima is obtained by taking any solution x; '"'(t), and running it backwards in time. It is easy to check that x; ' '(T — t) solves the Euler-Lagrange equations which follow from minimizing U in Eq. (7), and that the activation barrier U resulting from such a path is U=2b 7 ( ' Newt. )2 2 (9) Paths of this second type can start at the bottom of an energy well at t = 0, and evolve towards some higher energy at t = ~. It is therefore natural to guess that the best way to escape from a trapping potential is for a system to follow a path of this second type from t = 0 until some intermediate time v-&, at which point the system has reached the top of the barrier which restrains it. Now the system can switch over to a path of the first type, The argument of the exponential in Eq. (3) has a nice physical interpretation. It says that the most important thermal histories are those which minimize deviation from Newton's laws. I now assume that the functional integral in Eq. (3) is dominated by a single path which maximizes the integrand [8]. The approximation to be explored, therefore, is g(x, v: T) — e (6) where the activation barrier U(r) is 0031-9007/95/74(22)/4547(4)$06. 00 1995 The American Physical Society 4547