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Next: Accuracy and Stability Up: Continuous Valued Cellular Previous: Introduction

Fermi, Pasta, Ulam Wave Equation

We begin by describing the nonlinear string in the FPU model in more detail. When the string is at rest it lies along the x-axis. When the string moves it only moves in the y direction. The string is modeled by considering a series of particles along the x-axis with fixed x coordinates that are separated by a distance, (so it follows that the mass of each particle is , where is the constant density of the string). Each particle's motion in the y direction is modeled by considering the effect of springs connecting each particle to its two neighbors. In Rheology, long polymeric molecules are often modeled by considering these ``bead-spring" type models, where each bead/particle represents 10-30 monomer units. When the springs are assumed to follow Hooke's law, these macromolecular models are called Rouse chains; when the springs are not Hookean, they are called Rouse-Zimm chains. [6]

Let represent the vertical displacement (i.e., the value of y) at time of the particle whose horizontal location is , and let represent the difference in the vertical displacement between two neighboring particles. We can express the distance, , between two neighboring particles as . The magnitude of the force for most springs will usually be some function G of this distance (divided by so that the force is normalized), i.e.,

Since it is assumed that the particles do not move in the x direction, we are only concerned with the vertical component of the spring force. The absolute value of the force's vertical component is given by

which we can express strictly as a function of :

If the spring is linear (i.e., follows Hooke's law) and has an equilibrium length of z=0, as is the case for Rouse chains, then , where k is the positive-valued spring constant, and therefore . For linear springs with nonzero equilibrium lengths or nonlinear springs where is an analytic function near z=1, we can express as a series of even powers of when is small, which leads to the following expression for :

If the spring force is an analytic function of instead of an analytic function of distance, we may also have even powers in the series:

Now we consider the effect of the forces on a generic particle, located at . If we define and then we can express Newton's law by using eq (4b) to describe the effect of each of the two springs acting on the particle:

As approaches 0 in eq (5), we obtain the nonlinear partial differential equation corresponding to the FPU model:

We consider some special cases of our model. If the springs follow Hooke's law and have an equilibrium length of 0, then the nonlinear terms disappear and we are left with the standard linear wave equation

where , the spring constant. If stays small in the nonlinear model we can ignore the effects of higher order terms. This leads to the quadratic FPU equation

which FPU's scheme (ii) simulated. When the spring force is strictly a function of distance, we work with eq (4a) instead of eq (4b), so . For this common case, we obtain the cubic FPU equation when stays small

which FPU's scheme (iii) simulated. Eqs (7)-(9) will be the three equations of interest in this paper.

next up previous
Next: Accuracy and Stability Up: Continuous Valued Cellular Previous: Introduction

Rudy Rucker
Sat May 9 20:21:15 MET DST 1998