Fermi-Pasta-Ulam Model

The Fermi-Pasta-Ulam (FPU) model is a model for a one-dimensional collection of particles with massless, weakly anharmonic (nonlinear) springs connecting them to each other. Letting $\{q_j\}_{j=1}^{N}$ and $\{p_j\}_{j=1}^{N}$ denote the position and momentum coordinates of an $N$-particle chain, we can define the FPU model Hamiltonian:

$$H= \sum\limits_{j=1}^N \left( \frac{p_j^2}{2 m} + \kappa \fr... ...(q_j-q_{j+1})^3}{3}+ \beta \frac{(q_j-q_{j+1})^4}{4} \right)$$ (1)

Here we assumed periodic boundary conditions $p_{N+1}=p_1$ and $q_{N+1}=q_1$. Equivalently, the beads are connected in a circular arrangement. The parameter $m$ denotes the particle mass, while $\kappa$, $\alpha$, and $\beta$ are coefficients involving the spring properties.

The equations of motion are the standard Hamilton's equations:

$$\dot{q}_j=\frac{\partial{{\fam2 H}}}{\partial p_j}, \ \ \dot{p}_j=-\frac{\partial {{\fam2 H}}}{\partial q_j}$$ (2)

We will here focus on the $\alpha$-FPU model for which $\alpha \neq 0$ and the quartic term is absent ($\beta = 0$). We nondimensionalize the system with respect to the spring constant $\kappa$, the mass $m$, and the energy density $H/N$. Retaining the original symbols for the nondimensionalized variables $p_j$ and $q_j$, we obtain the nondimensionalized Hamiltonian and equations of motion:

$${\mathcal H} = \frac{1}{2}\sum\limits_{j=1}^N \left( {p_j^2} + (q_j-q_{j+1})^2\right)+\frac{\epsilon}{3} \sum\limits_{j=1}^N (q_j-q_{j+1})^3$$ (3)

$$\dot q_j = p_j$$

$$\dot p_j = \left(q_{j-1}-2 q_j + q_{j+1} \right)\left(1+\epsilon(q_{j-1}-q_{j+1})\right)$$

Our choice of nondimensionalization implies that

$${\mathcal H}= N$$ (4)

for all times. The fundamental nondimensional parameter measuring the strength of the nonlinearity is

$$\epsilon \equiv \alpha \sqrt{\frac{H}{N}}.$$

In order to study the transfer of energy among different scales, we represent the system in terms of Fourier modes:

$$\left(\begin{array}{c}q_l \\ p_l \end{array}\right) =\frac{1}{N}... ...ray}{c}Q_k \\ P_k \end{array}\right) \exp{\left(\frac{-2\pi i k l }{N}\right)}$$

$$\left(\begin{array}{c}Q_k \\ P_k \end{array}\right) = \sum\limit... ...rray}{c}q_l \\ p_l \end{array}\right) \exp{\left(\frac{2\pi i k l }{N}\right)}$$ (5)

The Hamiltonian in the new variables reads

$${\mathcal H} = \frac{1}{2N} \sum\limits_{k=-\frac{N}{2}+1}^{\frac{N}{2}} \left[ \vert P_k\vert^2+\omega_k^2 \vert Q_k\vert^2\right]$$ (6)

$$\qquad +\frac{\epsilon}{3N^{2}} \sum\limits_{k_1, k_2, k_3= -\fra... ...{\frac{N}{2}} V_{k_1, k_2, k_3} Q_{k_1} Q_{k_2} Q_{k_3} \delta_{k_1+k_2+k_3,0},$$

where the dispersion relation is given by

$$\omega_k=2 \left\vert\sin{\left(\frac{\pi k}{N}\right)}\right\vert,$$ (7)

the nonlinear coupling coefficients are

$$V_{k_1,k_2,k_3} = -i \,\mathrm{sgn}\,(k_1k_2k_3) \omega_{k_1} \omega_{k_2} \omega_{k_3}, \ \ \ k_1+k_2+k_3=0,$$ (8)

and

$$\delta_{i,j} \equiv \begin{cases}1 & \mbox{if } i = j \mod N, \\ 0 & \mbox{else.} \end{cases}$$

is a periodized version of the Kronecker delta function.

To quantify the amplitude of activity of the FPU chain at different scales, we define the harmonic energy contribution of each Fourier mode:

$$E_{h,k}(t) \equiv \frac{1}{2N} \left[ \vert P_k\vert^2+\omega_k^2 \vert Q_k\vert^2\right].$$

Energy equipartition implies $E_{h,k}$ is independent of $k$ (and $t$). The total harmonic contribution to the energy is

$$E_h(t) \equiv \sum_{k= - N/2+1}^{N/2} E_{h,k}(t).$$