For several years I taught an introductory graduate course
entitled * Methods of Statistical Physics* at the University of
Maryland that emphasizes ensemble methods for equilibrium statistical mechanics
and thermodynamics. Details of the syllabus can be found at the course web
page,

Although many good textbooks on ** Statistical Physics**
are available, I noticed while preparing my first set of lecture notes that
figures were often reproduced from one generation of textbook to the next with
decreasing fidelity, with the result that significant distortions or even
outright errors eventually appeared in some recent books. The problem
appears to be that creation of these figures often requires substantial
numerical effort that recent authors eschewed, relying instead upon illustrators
to produce faithful adaptations from classic texts. Unsatisfied with this
approach, I endeavored to reproduce many of these figures and found that
performing the numerical work was often quite instructive in itself. Not
being a specialist in statistical physics, I derived considerable satisfaction
from performing both the derivations and the computations needed to solve a wide
variety of problems and to produce accurate figures. Therefore, I decided
to provide my students with course materials that would help prepare them for
computational methods of problem solving. After all, first-year graduate
students are not specialists yet either and hopefully would appreciate achieving
self-sufficiency also.

Traditional physics education tends to limit its choice of problems to those which a clever mathematician can solve in symbolic form, but modern research depends upon computational tools because relatively few interesting problems can be solved using purely analytical techniques. Furthermore, even if a symbolic solution is possible, it is often very difficult to produce accurate sketches of the dependencies of various thermodynamic properties upon the relevant variables without the assistance of a computer program. Although such plots are essential to the development of understanding and physical insight, few students can produce realistic sketches unaided; in fact, most instructors reproduce on the blackboard figures produced by computers and imply that students should achieve comparable accuracy without such assistance.

* **Mathematica* is a powerful symbolic
manipulator which provides very useful tools for solving problems and exploring
the results. Its symbolic, numerical, and graphical tools allow the student to focus more
upon physics than upon either algebra or numerical algorithms. Although
custom programs using compiled languages like Fortran or C can provide faster
large-scale computation, I often develop the outline of a solution and numerical
algorithms using *Mathematica* first and then test my compiled programs
against it. The sooner that students become proficient with computer-aided
mathematics, using *Mathematica* or a related system, the more productive
researchers they will become. It may not be a panacea, but such tools are
essential to modern science. The present courseware for statistical
physics assumes that the reader is already familiar with *Mathematica*, but
we also offer a course entitled *Essential
Mathematica for Students of Science** *that teaches the fundamentals
of *Mathematica* using a tutorial approach that is suitable for self study;
the ** getting started** notebook provides sufficient background for
this course.

Two types of notebooks are provided. ** Formalism**
notebooks present the formalism of equilibrium statistical physics in a format
that is similar to that of traditional textbooks, except that hyperlinks are
often provided to related materials. These notebooks use the typesetting
features of

The formalism notebooks and many of the application
notebooks include many "end-of-chapter" problems. Some of these
problems can be solved by traditional means, but many require numerical
methods. Although some are purely formal in nature, most would benefit
from the computational or graphical capabilities offered by *Mathematica*.
Students are encouraged to submit well-formatted notebooks that utilize the
typesetting features of the system. Detailed solutions are available to
all problems assigned as homework in the last few years and can be supplied to
qualified instructors upon request. These
solutions have not been made public so that instructors can assign homework
without the solutions being in general circulation.

The notebooks may be accessed using
the hyperlinks below. To view the files directly, you must configure your
web browser to launch either *Mathematica* or *MathReader* for files
with the *nb* extension. *MathReader*
is a free program that
will display or print a *Mathematica *notebook, but it cannot evaluate or
edit notebooks.
Alternatively, you may download files to your computer and launch the
application manually.

Most of the application notebooks were originally written in 1996, but have been
extensively revised in subsequent years; please check the dates to make sure that your
copies are current. The present versions were prepared using *Mathematica
4.1,*
but most should still run under *Mathematica 3.0* with little or no
change. The formalism notebooks were written in 2000 and 2001 to provide a
more consistent presentation than was possible using textbooks with different
notational conventions for formal development of the theory. Although
several different traditional textbooks were chosen for the course in various
semesters, their usefulness has decreased as the present materials became more
complete.

*Last updated: June 25, 2002*

- ReviewThermodynamics
- Reviews thermodynamics and some of the techniques for
derivation of thermodynamic relationships.
*Revised May 11, 2002* - StatisticalPostulate
- The basic postulates of statistical mechanics are used to derive and
explain the laws of thermodynamics. Our approach relies upon the
information-theory concept of disorder and identifies the disorder within a
statistical ensemble with thermodynamic entropy.
*Revised May 11, 2002* - Ensembles
- Two methods for construction of canonical probability distributions are
presented. The first is based upon thermal interaction between a sample
and a much larger reservoir of heat. The second maximizes entropy
subject to constraints upon mean values of energy and perhaps other variables.
The microcanonical, canonical, and grand canonical ensembles are developed in
detail.
*Revised May 11, 2002* - Semiclassical
- The properties of ensembles composed of points in classical phase space are
studied. Two important theorems, equipartition and virial, are
developed. Correspondence with quantum mechanics is used to establish a
fundamental cell size in phase space that permits computation of finite
entropy for semiclassical ensembles. Applications are made to ideal
gases and diatomic molecules.
*Revised May 14 2002* - Fluids
- The effects of intermolecular interactions upon the mechanical and thermal
equations of state are studied for classical fluids. The temperature
dependence of the second virial coefficient, which governs these effects for
dilute systems, is derived for realistic potentials and explained using a
model from which one can also derive the van der Waals equation. Next we
discuss the measurement of the pair correlation function in denser systems
using X-ray or neutron scattering. Finally, the relationship between
correlations and density fluctuations is developed.
*Revised May 29 2002* - IdealQuantumGases
- The indistinguishability of identical particles has profound effects at low
temperatures and/or high densities where quantum mechanical wavepackets
overlap appreciably. The occupation representation is used to study the
statistical mechanics and thermodynamics of ideal quantum gases satisfying
Fermi-Dirac or Bose-Einstein statistics. This notebook concentrates upon
formal and conceptual developments, while the auxiliary notebooks
*occupy.nb*,*fermi.nb*, and*bose.nb*provide technical support and further developments.*Revised May 14, 2002*

- spin-half.nb
- Uses the microcanonical ensemble to investigate the paramagnetism of
spin-1/2 systems. The physical interpretation of the phenomenon of negative
spin temperature is discussed in some detail.
*Revised May 29, 2002* - hotherm.nb
- Uses the microcanonical ensemble to investigate the thermodynamics of
independent oscillators. The Einstein model of lattice vibrations is presented.
*Revised May 29, 2002* - ising1d.nb
- The combinatorial method is used to solve the Ising model for a
one-dimensional chain of spin-1/2 atoms in an external magnetic field with
nearest neighbor spin-spin interactions.
*Revised May 29 2002* - thermo2.nb
- Uses the canonical ensemble to investigate the thermodynamics of binary
systems. The Schottky effect is discussed as a manifestation of the
quantization of the excitation-energy spectrum.
*Revised May 29, 2002* - paramag.nb
- Investigates the thermodynamics of paramagnetism for arbitrary spin using
the canonical ensemble. The classical limit is developed also.
*Revised May 29, 2002* - Weiss.nb
- Uses a mean-field model to study spontaneous magnetization for ferromagnetic
systems, with particular attention to behavior near the critical point.

*Revised May 14 2002* - debye.nb
- Compares the Debye and Einstein models of lattice vibrations of a crystal.
The Grueneisen model of expansivity is also developed.
*Revised May 14, 2002* - planck.nb
- Compares the Planck model of black-radiation with earlier classical
models. Also discusses Hawking radiation from black holes.
*Revised May 14, 2002* - rotvib.nb
- Studies rotational and vibrational contributions to the heat capacity of
ideal gases composed of diatomic molecules.
*Revised May 29, 2002* - virial.nb
- The second virial coefficient for classical gases is evaluated for
realistic intermolecular potentials.
*Revised May 29, 2002* - vdwaals.nb
- Properties of the van der Waals gas, including the Maxwell construction,
are developed.
*Revised May 29, 2002* - occupy.nb
- Investigates the statistics of occupation numbers for the Fermi-Dirac,
Bose-Einstein, and Maxwell-Boltzmann distributions. The dependencies on both
temperature and chemical potential are evaluated.
*Revised May 14, 2002* - fermi.nb
- Investigates the thermodynamics of ideal nonrelativistic Fermi-Dirac
gases.
*Revised May 14, 2002* - bose.nb
- Investigates the thermodynamics of ideal nonrelativistic Bose-Einstein
gases.
*Revised May 14, 2002*

Please notify me of any errors you find. Comments
and suggestions are welcome. My e-mail address is *jjkelly@physics.umd.edu*.