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In this simulation the bonding is done with a Lennard-Jones (LJ) potential. LJ potentials have extremely large repelling forces when the particles get too close and longer range attractive forces when the particles are farther apart. The LJ potential is of the form

`V(r)=4V_(LJ)[(sigma/r)^2b-(sigma/r)^b]`

where `V_(LJ)` is the potential minimum, `sigma` is a length scaling parameter, `r` is the particle separation, and `b` is an integer that is adjustable by the learner. The larger the value of `b`, the more abrupt is the force change with distance, r. The solid is set up so that all the particles are separated by a distance that causes the nearest neighbor forces to be zero. The force between a site and its second nearest neighbor is not initially zero and this effect can easily be seen when starting with no excited columns. An obvious result of that set up is that alternate rows of particles have to be staggered by 1/2 the spacing in the horizontal direction and the vertical spacing of rows is `sqrt(3)/2` times the horizontal spacing of particles in any column. The initial configuration is that of a perfectly regular crystal. Some of the left side (learner adjustable) columns have non-zero initial kinetic energy and remaining columns have zero initial kinetic energy. The object of the animation is to view how the energy spreads to the remaining columns and determine some time constant for the energy to become uniform in the `x` direction. A lattice site position represents the location of the nucleus of the atom, some of which are initially made to vibrate is random directions. It is important to understand that energy of a given lattice site can quickly (even in a single numerical iteration) spread to site's nearest neighbors because of the proximity of the sites and the strong potentials bonding the sites. Since we are limited to a very finite number of lattice sites, about the best we can hope to analyze is how the position of the centroid of the energy(x) distribution changes as the energy(x) becomes uniform. Therefore the mean coordinate, `x_(avg)(E)`, of the column energy is plotted Vs time in blue. In addition, I have provided a least squares exponential fit of `x_(avg)(E)` which makes an estimate of the energy equilibration time. If we denote that horizontal length of the lattice as L and x=0 at the center then we would expect that the energy centroid`x_(avg)(E)=(int_0^LxE(x)dx)/(int_0^LE(x)dx)`

should go to `L/2` as the energy becomes equilibrated with time and that is what we see with the time constant computed by the exponential fit `(x_(avg)(E))/L` Vs time. One feature of the `(x_(avg)(E))/L` plot is that it oscillates sinusoidally. The period of this oscillation is just the period of a particle that stays near the bottom of the Lennard Jones energy well. The rate of heat flow is roughly proportional to the frequency of this oscillation.This time dependent energy movement problem would normally be solved by the diffusion equation

`C(dT)/dt=d/dx(K(dT)/dx)`

where T is the temperature (the kinetic energy here), C is the heat capacity and K is the heat conductivity of the medium. My animation shows that the diffusion equation does not provide enough detail for such a microscopic lattice as this. I have also provided a 1 dimensional solution to the energy flow diffusion equation. Let `e` be the symbol for energy density, which in iD is Joules/meter. Then the energy flow diffusion equation is`(de)/dt=d/dx(K(de)/dx)`

where e is the 1D energy density (Joules/meter) and `(de)/dt` has units of watts per meter and since `(d^2e)/(dx^2)` has units Joules/meter^3, and K has units of (watts-meter^2)/Joule.

The diffusion equation is solved numerically and plotted (all plots are color coded to their labels) in the same area as the numerical solution Vs time and position.

The learner may adjust K to make the diffusion equation conduction speed agree with the numerical model of the solid. It is important to be sure that total energy is conserved when integrating the Lennard-Jones equations of motion. The lower plot shows that the total energy (potential+kinetic) is much more stable than the tradeoff between kinetic and potential energy.

Heat flow physics in solids can be separated into two very different mechanisms. First is a non-conductor like we have here, where there is no motion of the heat carrying media and second is a conductor which has very fast moving electrons to carry energy to many sites quickly. Conductors usually have conductivities, K, about 100 times greater than non-conductors. In fact, it is mainly the thermal motion of the lattice sites that limit the conductivity of a conductor type material so that a low temperature conductor's conductivity is much greater.

**I have deliberately allowed slider variable ranges that can result of a break up of
the solid. It that happens, the learner just needs to refresh the page, which in Windows means hittint the F5 key.**