Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2015
8.333: Statistical Mechanics I
Solutions to Problem Set # 2
Fall 2015
Probability
1. Random deposition: A mirror is plated by evaporating a gold electrode in vaccum by
passing an electric current. The gold atoms fly off in all directions, and a portion of
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
I. Thermodynamics
I.A Fundamental definitions
Thermodynamics is a phenomenological description of equilibrium properties of macroscopic systems.
As a phenomenological description, it is based on a number of empirical observations
which are summarized by
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
I.D The Second Law
The historical development of thermodynamics follows the industrial revolution in the
19
th
century, and the advent of heat engines. It is interesting to see how such practical
considerations as the efficiency of engines can lead to abs
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
III.E The HTheorem and Irreversibility
The second question posed at the beginning of this chapter was whether a collection
of particles naturally evolves towards an equilibrium state. While it is possible to obtain
steady state solutions for the full phas
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
II. Probability
II.A General Definitions
The laws of thermodynamics are based on observations of macroscopic bodies, and
encapsulate their thermal properties. On the other hand, matter is composed of atoms
and molecules whose motions are governed by funda
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
I.G Approach to Equilibrium and Thermodynamic Potentials
Evolution of nonequilibrium systems towards equilibrium is governed by the second
law of thermodynamics. For example, in the previous section we showed that for an adiabatically isolated system ent
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
II.D Many Random Variables
With more than one random variable, the set of outcomes is an N dimensional space,
S x = cfw_ < x1 , x2 , , xN < . For example, describing the location and velocity of a
gas particle requires six coordinates.
The joint PDF p(x
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
III.C The BogoliubovBornGreenKirkwoodYvon Hierarchy
The full phase space density contains much more information than necessary for description of equilibrium properties. For example, knowledge of the one particle distribution
is sufficient for computi
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
III. Kinetic Theory of Gases
III.A General Definitions
Kinetic theory studies the macroscopic properties of large numbers of particles, starting from their (classical) equations of motion.
Thermodynamics describes the equilibrium behavior of macroscopic
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
III.G Conservation Laws
Approach to equilibrium: We now address the third question posed in the introduction,
of how the gas reaches its final equilibrium. Consider a situation in which the gas is
perturbed from the equilibrium form described by eq.(III.
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
I.I Stability Conditions
The conditions derived in section I.G are similar to the well known requirements for
mechanical stability. A particle moving in an external potential U settles to a stable
equilibrium at a minimum value of U . In addition to the v
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2015
8.333: Statistical Mechanics I
Solutions to Problem Set # 1
Fall 2015
Thermodynamics
1. NonCarnot Engine: Consider an engine that operates between a set of temperatures
Tmax = T1 > T2 > T3 > > Tn = Tmin . For a subset of these temperatures cfw_T+ ,
the e
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
I I I.G Conservation Laws
Approach to equilibrium: We now address the third question posed in the intro duction,
of how the gas reaches its nal equilibrium. Consider a situation in which the gas is
perturbed from the equilibrium form described by eq.(I I
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
I.G Approach to Equilibrium and Thermo dynamic Potentials
Evolution of nonequilibrium systems towards equilibrium is governed by the second
law of thermo dynamics. For example, in the previous section we showed that for an adia
batically isolated system
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
I. Thermodynamics
I.A Fundamental denitions
Thermodynamics is a phenomenological description of equilibrium properties of macro
scopic systems.
As a phenomenological description, it is based on a number of empirical observations
which are summarized by
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
I I.D Many Random Variables
With more than one random variable, the set of outcomes is an N dimensional space,
S x = cfw_ < x1 , x2 , , xN < . For example, describing the lo cation and velo city of a
gas particle requires six co ordinates.
The joint PDF
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
I.D The Second Law
The historical development of thermo dynamics follows the industrial revolution in the
19
th
century, and the advent of heat engines. It is interesting to see how such practical
considerations as the eciency of engines can lead to abstr
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
I I I.C The Bogoliub ovBornGreenKirkwo o dYvon Hierarchy
The full phase space density contains much more information than necessary for de
scription of equilibrium properties. For example, knowledge of the one particle distribution
is sucient for comp
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
III. Kinetic Theory of Gases
I I I.A General Denitions
Kinetic theory studies the macroscopic properties of large numbers of particles, start
ing from their (classical) equations of motion.
Thermo dynamics describes the equilibrium behavior of macroscopi
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
I.I Stability Conditions
The conditions derived in section I.G are similar to the well known requirements for
mechanical stability. A particle moving in an external potential U settles to a stable
equilibrium at a minimum value of U . In addition to the v
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
II. Probability
I I.A General Denitions
The laws of thermo dynamics are based on observations of macroscopic bodies, and
encapsulate their thermal properties. On the other hand, matter is composed of atoms
and molecules whose motions are governed by funda
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
8.333: Statistical Mechanics I
Fall 2007
Test 3
Review Problems & Solutions
The third inclass test will take place on Wednesday 11/28/07 from
2:30 to 4:00 pm. There will be a recitation with test review on Monday 11/26/07.
The test is closed bo ok, but i
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
8.333: Statistical Mechanics I
Fall 2007
Test 2
Review Problems
The second inclass test will take place on Wednesday 10/24/07 from
2:30 to 4:00 pm. There will be a recitation with test review on Monday 10/22/07.
The test is closed bo ok, and composed ent
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
8.333: Statistical Mechanics I
Fall 2007
Test 1
Review Problems
The rst inclass test will take place on Wednesday 9/26/07 from
2:30 to 4:00 pm. There will be a recitation with test review on Friday 9/21/07.
The test is closed bo ok, but if you wish you m
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
I I I.E The HTheorem and Irreversibility
The second question posed at the beginning of this chapter was whether a collection
of particles naturally evolves towards an equilibrium state. While it is possible to obtain
steady state solutions for the full ph
Statistical Mechanics 1: Statistical Mechanics of Particles
PHYS 8.333

Fall 2007
8.333: Statistical Mechanics I
Re: 2007 Final Exam
Review Problems
The enclosed exams (and solutions) from the previous years are intended to help you
review the material.
*
Note that the rst parts of each problem are easier than its last parts. Therefore