Kardar statistical physics particles solutions download

Division by N! The factor of 2N takes into account the two possible signs for each pi. Since we are not interested in the coordinates, we can get the probability from the ratio of phase spaces for the momenta, i. Hard sphere gas: Consider a gas of N hard spheres in a box. There are no other interactions between the spheres, except for the constraints of hard-core exclusion. The joint integral over the spacial coordinates with excluded volume constraints is best performed by introducing particles one at a time.

Note that the joint effective excluded volume that appears in the above expressions is one half of the total volume excluded by N particles. Non-harmonic gas: Let us reexamine the generalized ideal gas introduced in the previous section, using statistical mechanics rather than kinetic theory.

We assume that A and s are both real and positive. Note how the usual equipartition theorem is modified for non-quadratic degrees of freedom. This evaluates to 32 kB T for the 3—dimensional ideal gas. The gas in part c is ideal in the sense that there are no molecule—molecule interactions. Surfactant adsorption: A dilute solution of surfactants can be regarded as an ideal three dimensional gas. As surfactant molecules can reduce their energy by contact with air, a fraction of them migrate to the surface where they can be treated as a two dimensional ideal gas.

Surfactants are similarly adsorbed by other porous media such as polymers and gels with an affinity for them. A c Gels are formed by cross-linking linear polymers. It has been suggested that the porous gel should be regarded as fractal, and the surfactants adsorbed on its surface treated as a gas in df dimensional space, with a non-integer df. Can this assertion be tested by comparing the relative adsorption of surfactants to a gel, and to the individual polymers presumably one dimensional before cross-linking, as a function of temperature?

Thus by studying the adsorption of particles as a function of temperature one can determine the fractal dimensionality, df , of the surface.

The largest contribution comes from the difference in energies. Molecular adsorption: N diatomic molecules are stuck on a metal surface of square symmetry. Each molecule can either lie flat on the surface in which case it must be aligned to one of two directions, x and y, or it can stand up along the z direction. What is the largest microstate energy? Note that the ensemble corresponding to the macrostate T, B includes magnetic work.

Langmuir isotherms: An ideal gas of particles is in contact with the surface of a catalyst. Find P0 T. S z is quantized to -1, 0, Ignore all other degrees of freedom. The contribution of the rotational degrees of freedom to the Hamiltonian is given by Hrot. Long range interactions can result in non-analytic corrections to the ideal gas equation of state. The spinodal line indicates onset of metastability and hysteresis effects.

In the interval between the two curves, the system is locally stable, but globally unstable. The formation of ordered regions in this regime requires nucleation, and is very slow. The dashed area is locally unstable, and the system easily phase separates to regions rich in A and B. The first and last particles are held fixed at the equilibrium separation of N a.

In a normal mode, the particles oscillate in phase. The usual procedure is to obtain the modes, and corresponding frequencies, by diagonalizing the matrix of coefficeints coupling the displacements on the right hand side of the equation of motion. This is a consequence of translational symmetry, and allows us to diagonalize the matrix using Fourier modes. The number of normal modes thus equals the number of original displacement variables, as required.

Furthermore, the amplitudes are chosen such that the normal modes are also orthonormal, i. The next step is to change the coordinates of phase space from uj to an. Plot the resulting squared displacement of each particle as a function of its equilibrium position.

Find the relationship between the radius and mass of a black hole by setting this escape velocity to the speed of light c. Relativistic calculations do not modify this result which was originally obtained by Laplace. Consider the coalescence of two solar mass black holes. Find the temperature of the black hole in terms of its mass. Find the rate of energy loss due to such radiation.

Near the boundary of a black hole, sometimes one member of a pair falls into the black hole while the other escapes. This is a hand-waving explanation for the emission of radiation from black holes. How long is this time for a black hole of solar mass? According to the recently formulated Holographic Principle there is a maximum to the amount of entropy that this volume of space can have, independent of its contents! What is this maximal entropy? This course is the first part of a two-course sequence.

The sequence continues in 8. Mehran Kardar. Fall For more information about using these materials and the Creative Commons license, see our Terms of Use.

Instructor s Prof. Some Description Instructor s Prof. Viscosity: Consider a classical gas between two plates separated by a distance w. Light and matter: In this problem we use kinetic theory to explore the equilibrium between atoms and radiation. What does this imply about the above cross sections? What is the penetration length across which the incoming flux decays?

Hint: Use the method of Lagrange multipliers to impose the constraint. Hence show that the kinetic energy per particle can serve as an empirical temperature. Moments of momentum: Consider a gas of N classical particles of mass m in thermal equilibrium at a temperature T , in a box of volume V.

Generalized ideal gas: Consider a gas of N particles confined to a box of volume V in d-dimensions. By examining the s E net force on an element of area prove that the pressure P equals d.

V , where E is the average kinetic energy. They cannot climb the wall, but can escape through an opening of size 5mm in the wall. Effusion: A box contains a perfect gas at temperature T and density n. A small hole is opened in the wall of the box for a short time to allow some particles to escape into a previously empty container.

Ignore the possibility of any particles returning to the box. Hint: calculate contributions to kinetic energy of velocity components parallel and perpendicular to the wall separately. What is the final temperature of the gas in the container? After 30 days it is found that 24mg of mercury has been lost. What is the vapor pressure of mercury at 00 C? Adsorbed particles: Consider a gas of classical particles of mass m in thermal equilib- rium at a temperature T , and with a density n.

A clean metal surface is introduced into the gas. Particles hitting this surface with normal velocity less than vt are reflected back into the gas, while particles with normal velocity greater than vt are absorbed by it. Electron emission: When a metal is heated in vacuum, electrons are emitted from its surface. Hint: By appropriate change of scale, the surface of constant energy can be deformed into a sphere.

A more elegant method is to implement this deformation through a canonical transformation. The surface of constant energy is an ellipsoid in 2N dimensions, whose area is difficult to calculate.

Hence calculate the mean kinetic energy, and mean potential energy for each oscillator. E p21! Consider a microcanonical ensemble of total energy E.

Division by N! Since we are not interested in the coordinates, we can get the probability from the ratio of phase spaces for the momenta, i. Hard sphere gas: Consider a gas of N hard spheres in a box. There are no other interactions between the spheres, except for the constraints of hard-core exclusion.

N 2 3N h2 b Calculate the equation of state of this gas. Non-harmonic gas: Let us reexamine the generalized ideal gas introduced in the previous section, using statistical mechanics rather than kinetic theory.

We assume that A and s are both real and positive. Note how the usual equipar- tition theorem is modified for non-quadratic degrees of freedom. This evaluates to 32 kB T for the 3—dimensional ideal gas. The gas in part c is ideal in the sense that there are no molecule—molecule interactions.

Surfactant adsorption: A dilute solution of surfactants can be regarded as an ideal three dimensional gas. As surfactant molecules can reduce their energy by contact with air, a fraction of them migrate to the surface where they can be treated as a two dimensional ideal gas.

Surfactants are similarly adsorbed by other porous media such as polymers and gels with an affinity for them. A c Gels are formed by cross-linking linear polymers. It has been suggested that the porous gel should be regarded as fractal, and the surfactants adsorbed on its surface treated as a gas in df dimensional space, with a non-integer df. Can this assertion be tested by comparing the relative adsorption of surfactants to a gel, and to the individual polymers presumably one dimensional before cross-linking, as a function of temperature?

Thus by studying the adsorption of particles as a function of temperature one can determine the fractal dimensionality, df , of the surface. Molecular adsorption: N diatomic molecules are stuck on a metal surface of square symmetry. Each molecule can either lie flat on the surface in which case it must be aligned to one of two directions, x and y, or it can stand up along the z direction. What is the largest microstate energy? Note that the ensemble corresponding to the macrostate T, B includes magnetic work.

Langmuir isotherms: An ideal gas of particles is in contact with the surface of a catalyst. Find P0 T. S z is quantized to -1, 0, Ignore all other degrees of freedom. The contribution of the rotational degrees of freedom to the Hamiltonian is given by! Long range interactions can result in non-analytic corrections to the ideal gas equation of state. This overcounts by the number of equivalent arrangements that appear in the denom- inator.

Hence N! This is due to the long range nature of the Coulomb interaction. This is equivalent to an expectation value that can be calculated perturbatively in a cumulant expansion. How many pairs have repulsive interactions, and how many have attractive interactions? For like charges, we can choose pairs from the N particles of positive charge, or from the N particles with negative charges. Hence the number of pairs of like pairs is!

The appearance of two length scales a and L, makes the scaling analysis of part c questionable. What are the phases of this system at low and high temperatures? The partition function for two particles i. Thus the unphysical collapse at low temperatures is preempted at the higher temperature where the hard cores become important. Exact solutions for a one dimensional gas: In statistical mechanics, there are very few systems of interacting particles that can be solved exactly.

Such exact solutions are very important as they provide a check for the reliability of various approximations. A one dimensional gas with short-range interactions is one such solvable case.

Note that there is no N! By contrast, the approximate van der Waals equation or the mean-field treatment incorrectly predicts such a transition. The Manning transition: When ionic polymers polyelectrolytes such as DNA are immersed in water, the negatively charged counter-ions go into solution, leaving behind a positively charged polymer.

Because of the electrostatic repulsion of the charges left behind, the polymer stretches out into a cylinder of radius a, as illustrated in the figure. While thermal fluctuations tend to make the ions wander about in the solvent, electrostatic attractions favor their return and condensation on the polymer.

The average position is then Z R 2 2! Identify the transition temperature, and characterize the nature of the two phases. In particular, how does hri depend on R and a in each case? This is a qualitative question for which no new calculations are needed. Hard rods: A collection of N asymmetric molecules in two dimensions may be modeled as a gas of rods, each of length 2l and lying in a plane. A rod can move by translation of its center of mass and rotation about latter, as long as it does not encounter another rod.

You may ignore the momentum contributions throughout, and consider the large N limit. Can you identify a phase transition as the density is decreased?

Draw the corresponding critical density nc on your sketch. Surfactant condensation: N surfactant molecules are added to the surface of water over an area A. This simple form ignores the couplings to the fluid itself. The actual kientic and potential energies are more complicated.

Statistical Physics of Particles by Mehran Kardar. This book tsatistical not yet featured on Listopia. This course is the first part of a two-course sequence. Course Collections See related courses in the following collections: African Studies Review is the principal academic and scholarly journal of the African Studies Association.

Table of Contents 1. User Review — Flag as inappropriate Concise, elegant formulation all over.