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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:
\begin{displaymath}
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)
\end{displaymath} (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:

\begin{displaymath}
\dot{q}_j=\frac{\partial{{\fam2 H}}}{\partial p_j}, \ \
\dot{p}_j=-\frac{\partial {{\fam2 H}}}{\partial q_j}
\end{displaymath} (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:
$\displaystyle {\mathcal H}$ $\textstyle =$ $\displaystyle \frac{1}{2}\sum\limits_{j=1}^N \left( {p_j^2} +
(q_j-q_{j+1})^2\right)+\frac{\epsilon}{3}
\sum\limits_{j=1}^N
(q_j-q_{j+1})^3$ (3)
$\displaystyle \dot q_j$ $\textstyle =$ $\displaystyle p_j$  
$\displaystyle \dot p_j$ $\textstyle =$ $\displaystyle \left(q_{j-1}-2 q_j + q_{j+1} \right)\left(1+\epsilon(q_{j-1}-q_{j+1})\right)$  

Our choice of nondimensionalization implies that
\begin{displaymath}
{\mathcal H}= N
\end{displaymath} (4)

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

\begin{displaymath}
\epsilon \equiv \alpha \sqrt{\frac{H}{N}}.
\end{displaymath}

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

$\displaystyle \left(\begin{array}{c}q_l \\  p_l \end{array}\right)
=\frac{1}{N}...
...ray}{c}Q_k \\  P_k \end{array}\right)
\exp{\left(\frac{-2\pi i k l }{N}\right)}$      
$\displaystyle \left(\begin{array}{c}Q_k \\  P_k \end{array}\right)
=
\sum\limit...
...rray}{c}q_l \\  p_l \end{array}\right)
\exp{\left(\frac{2\pi i k l }{N}\right)}$     (5)

The Hamiltonian in the new variables reads

$\displaystyle {\mathcal H}$ $\textstyle =$ $\displaystyle \frac{1}{2N}
\sum\limits_{k=-\frac{N}{2}+1}^{\frac{N}{2}}
\left[ \vert P_k\vert^2+\omega_k^2 \vert Q_k\vert^2\right]$ (6)
    $\displaystyle \qquad +\frac{\epsilon}{3N^{2}}
\sum\limits_{k_1, k_2, k_3= -\fra...
...{\frac{N}{2}}
V_{k_1, k_2, k_3}
Q_{k_1} Q_{k_2} Q_{k_3}
\delta_{k_1+k_2+k_3,0},$  

where the dispersion relation is given by
\begin{displaymath}
\omega_k=2 \left\vert\sin{\left(\frac{\pi k}{N}\right)}\right\vert,
\end{displaymath} (7)

the nonlinear coupling coefficients are
\begin{displaymath}
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,
\end{displaymath} (8)

and

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

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:

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

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

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


next up previous
Next: Numerical Simulation of Relaxation Up: Stages of Energy Transfer Previous: Introduction
Dr Yuri V Lvov 2007-01-17