[MUSIC] When we try to classify a matter or particles, it's very useful to think of how they carry their energy. All right now, in a very simplistic way, a particle can carry its energy in motion. That is the kinetic energy, or in its position, the potential energy. Let's assume that we have a billion particles in a box, and the box is connected to a thermostat that holds it at a certain temperature. Now, there is a complicated set of interactions between all the particles inside the box. Now a simple way to start thinking about materials is to say that either the kinetic energy dominates or the potential energy dominates. Now what does this mean? Well if the kinetic energy dominates the particles are all jiggling around and the particles are moving around everywhere with no particular preference for any position. Now in this case we get a gas. What happens if the potential energy dominates? This means that the particles are not moving so much and they actually want to arrange themselves in a special configuration, a lattice that minimizes their energy, and we call this a solid. Now this simplified picture is correct in essence. Liquid on the other hand, is highly complicated. With liquids from this simple point of view, it's neither the kinetic energy nor the potential energy that dominates. Now in this lecture, we will aim to illustrate how to think about a classic liquid. Now at the end of this lecture you should be able to develop a model for a classical fluid, by summation of particle positions and momentum. You should also be able to derive how the pair distribution function, can be measured using an experimental technique, called x-ray diffraction. Now the thermodynamic behavior of any system is governed by its partition function. The partition function is simply a summation over all possible states. Now if we consider a system of molecules, this includes all the particles that comprise the molecule, that is, all the protons, the neutrons and electrons. Now, in most cases, the motion of the electrons can be averaged over all the quantum fluctuations leading to an effective interaction between the nuclei. For example, one extremely useful model to describe the interaction between molecules is the description written down in 1924 by John Lennard-Jones. The Lennard-Jones interaction potential as it is called is constructed by including two off setting physical contributions. For short inter-nuclei spacing, the Pauli exclusion principle dictates that energy diverges exponentially as the distance between the nuclei tend to zero. Now in the Lennard-Jones potential, this is approximated as an inverse power law that scales as the 12th power of the inter-nuclei spacing. Now for large inter-nuclei distances, the electronic polarize ability of the neutral atom leads to effective interaction that scales as one over r to the six. Now these forces are called Vandervalls forces and the Lennard-Jones potential is simply given as the sum of these two contributions. Now in this expression v nought is simply a potential scaling term. And sigma is a length scale of the interactions related to the polarizability of the items. Now the simple module ignores the coupling between the electron and the nuclei kinetic energies. This approximations are valid in the limit that the ratio of the electron mass to the nuclei mass ends to zero. Now when integrating over all the electronic motion, we're left with a simpler set of effective particles that interact with each other via an interaction potential. And this gives us a way to define a classical model for these interactions. Now lets consider a simple system, an n-particle system. Now in this classical picture, the degrees of freedom associated with the system are the coordinate positions and the momentum of the particles. The microscopic state is defined by a point in the face base. Now this face base includes all the momentum and all the positions. Now, by definition, the energy of a point in this face base is simply given by the Hamiltonian H. Now the Hamiltonian, remember, is simply a sum of the total kinetic energy and the total potential energies. The points in a face space form a continuum. So the classical partition function can be changed from a summation and converted into an integral that is given by. Now in this relation A is a factor that is used to ensure that the partition function is dimensionless. Now the factor A actually corrects for the approximation that we have made that the face space a continuum. Now this parameter A is universal and it actually does not depend on the features of the Hamiltonian H. Now how we can we evaluate this universal factor A? Given this is universal it must hold for all systems. Now the simplest system where we can evaluate the partition function exactly is the ideal gas. Now in the case of an ideal gas the molecules do not feel the effect of other molecules and as a result, the interaction potential is zero. Hence, the classical partition function for each particle is simply an integral over its momentum in the three directions scaled by the volume of the system. Now, each particle is non interacting. And hence the partition function can simply written as the single particle partition function raised to the power n where n is the total number of particles. Now remember, the partition function for the translational energy for an ideal gas is given by the following relation. Now this equality gives us a way to evaluate this universal factor A. Now the classical partition function can now be written as separating the kinetic and the potential energy parts of the Hamiltonian we then get. A new quantity emerges from this analysis. Which is the sum of the position phase base. And is typically noted by z and we call this the configuration of partition function. Thus once the configurational partition function is known everything about the classical partition function is. Now what is the probability distribution that governs the system? Now the probability of finding the system at location between R and R plus DR and momenta P and P plus an infinitely small change, DP, is given by the following equation. Now in order to the momentum distribution of any particle say a particle with index one then we can simply evaluate it by integrating over all the other particles. This yields the famous Maxwell Distribution which gives a way to determine the probability of finding a particle with speed between V and V+DV. Now from this distribution, we can easily evaluate what the most probable speed will be. We can now ask the question, well how do the energies of the molecules partition themselves? The very important equi partition theorem dictates this. This says that each terminally active degree of freedom is partitioned a fixed amount of energy. Now with the analysis that we have developed, is it possible to rationalize this equal partition thereon? Now the Hamiltonian of the system can be diagnolized into M degrees of freedom using a generalized coordinate system. Note that in the generalized coordinate system, each generalized coordinate is a superposition of the coordinates of the individual particles. Invoking this diagonalization gives us a way to write the Hamiltonian of the system as. Now, using this partition function we can evaluate the average energy associated with any particular degree of freedom. Now let's evaluate this for the momentum and so the average kinetic energy is simply the first moment of kinetic energy. Now this simply gives us a constant amount of energy given by one half times the Boltzmann constant, times the temperature. A similar analysis can be done for the normal modes, and those yield the same fixed amount of energy. Hence the average energy for a classical system is given by the sum of the degrees of freedom. Now what is this for an ideal gas? And what is it for a solid? Now when we have a monatomic ideal gas, now m takes the value 0 and n takes the value of 3n where 3n is the number of degrees of freedom for the translational energy. Hence all the energy is simply going to contain in the kinetic energy. And this gives the well-known average energy for a monatomic gas of 3 pi 2 n k3. Now on the other hand for a solid, in addition to the kinetic energy, there's an energy contained in the vibrations. Hence this leads to both m and n being equal to 3n. Where n is the number of particles. Now this leads to the average energy of a solid that is twice as much as that of a monatomic ideal gas.