Classical Statistical Mechanics in the Canonical Ensemble

Classical Statistical Mechanics 1. The Equipartition Theorem 2. The Classical Ideal Gas a. Kinetic Theory b. Maxwell-Boltzmann Distribution

The Equipartition Theorem Valid in Classical Statistical Mechanics ONLY!!! “Each degree of freedom in a system of particles contributes (½)k B T to the thermal average energy of the system.” Note: 1. This is valid only if each term in the classical energy is proportional either a momentum squared or a coordinate squared. 2. The possible degrees of freedom are those associated with translation, rotation &vibration of the system’s molecules.

Classical Ideal Gas For this system, it’s easy to show that The Temperature is related to the average kinetic energy. For one molecule moving with velocity v, in 3 dimensions this takes the form: Further, for each degree of freedom, it can be shown that

The Boltzmann Distribution: Define The Energy Distribution Function (Number Density) n V (E): This is defined so that n V (E) dE  the number of molecules per unit volume with energy between E and E + dE. The Canonical Probability Function P(E): This is defined so that P(E) dE  the probability to find a particular molecule between E and E + dE Z

Equipartition Simple Harmonic Oscillator Free Particle Z

Thermal Averaged Values Average Energy: Average Velocity: Of course:

Kinetic Theory of Gases & The Equipartition Theorem

Classical Kinetic Theory Results The kinetic energy of individual particles is related to the gas temperature as: (½)mv 2 = ( 3 / 2 ) k B T Here, v is the thermal average velocity. There is a wide range of energies (& speeds) that varies with temperature: Boltzmann Distribution of Energy

The Kinetic Molecular Model for Ideal Gases The gas consists of large number of small individual particles with negligible size. Particles are in constant random motion & collisions. No forces are exerted between molecules. From the Equipartition Theorem, The Gas Kinetic Energy is Proportional to the Temperature in Kelvin.

Maxwell-Boltzmann Velocity Distribution The Canonical Ensemble gives a distribution of molecules in terms of Speed/Velocity, & Energy. The One-Dimensional Velocity Distribution in the x-direction (u x ) has the form:

Low T High T

Maxwell-Boltzmann Distribution 3D Velocity Distribution: a  (½)[m/(k B T)] In Cartesian Coordinates:

Change to spherical coordinates: Reshape the box into a sphere in velocity space of the same volume with radius u. V = (4/3)  u 3 with u 2 = u x 2 + u y 2 + u z 2 dV = du x du y du z = 4  u 2 du Maxwell-Boltzmann Speed Distribution

3D Maxwell-Boltzmann Speed Distribution Low T High T

3D Maxwell-Boltzmann Speed Distribution Convert the velocity-distribution into an energy-distribution:  = (½)mu 2, d  = mu du

Velocity Values from the M-B Distribution u rms = root mean square velocity u avg = average speed u mp = most probable velocity

Comparison of Velocity Values Ratio in Terms of : u rms u avg u mp

Maxwell-Boltzmann Velocity Distribution

Maxwell-Boltzmann Speed Distribution

The Probability Density Function The random motions of the molecules can be characterized by a probability distribution function. Since the velocity directions are uniformly distributed, we can reduce the problem to a speed distribution function which is isotropic. Let f(v)dv be the fractional number of molecules in the speed range from v to v + dv. A probability distribution function has to satisfy the condition We can then use the distribution function to compute the average behavior of the molecules:

Some Other Examples of the Equipartion Theorem LC Circuit: Harmonic Oscillator: Free Particle in 3 D: Rotating Rigid Body :