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 = T
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
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 n
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
cons
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 poss
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
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
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
ga
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
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.
Thermodyna
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
pe
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
equ
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
p
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 sectio
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 nu
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
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
con
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, knowled
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 dy
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
equ
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
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 rev
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 Mond
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
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 po
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