2 On the Foundational Principles of Statistical Mechanics
91
the proof that the two-body potential used by Lee and Yang does not meet the most
important interactions in physics such as the electromagnetic interaction [12].
As mentioned above, in most standard textbooks of statistical mechanics in the
definition of the partition function (2.4) no region for integration, or the supporting set
of the distribution function, is specified. When ergodicity is broken, as discussed in
the last section, the region in the phase space visited by a physical system changes,
and the supporting set of the distribution function changes correspondingly. The
determination of the supporting set depends on the physical problem under study.
For example, when dealing with a crystal, one has to consider firstly the symmetry
of the lattice, which then specifies the supporting set of the distribution function.
Gibbs introduced the grand canonical ensemble to deal with chemical reaction
systems. As an example, we may consider the reaction H + H → H 2 of the combination of two hydrogen atoms into a hydrogen molecule to compare quantum
mechanics with statistical mechanics. A treatment in quantum mechanics starts from
the Hamiltonian of two hydrogen atoms. The Born-Oppenheimer approximation is
used to separate out the degrees of freedom for nuclei, and the Schrödinger equation
of electrons is solved for electronic levels U i (R) as a function of the nuclear distance R. Bounded states are then interpreted as a hydrogen molecule. However, in
the treatment of statistical mechanics, one has to include both the hydrogen atoms
and the hydrogen molecules in the grand canonical ensemble at the beginning. From
a pure atomic Hamiltonian no hydrogen molecules would be derived from statistical mechanics. The phase space for a pure atomic system of hydrogen atoms is
different from that for a mixture of hydrogen atoms and molecules with regard to
the breaking of ergodicity. The first order phase transition may be regarded as the
simplest chemical reaction A → B, and thus should be treated using grand canonical
ensemble.
Whether in a simulation of molecular dynamics or in a Monte-Carlo simulation a
prerequisite is that no change is made in the supporting set of distribution function.
When applying accelerated or enhanced means for sampling, we must be careful.
This is often overlooked in the literature.
2.6 Analogue to Thermodynamics: Heat and Free Energy
Thermodynamic is a branch of physics studying the relation between heat and temperature and between energy and work. It defines macroscopic variables such as
internal energy, entropy and pressure, and describes universal relations among them
without regard to any specific property of specific matter. Such general rules are
represented by the four laws of thermodynamics.
The early motivation for thermodynamics was a desire to increase the efficiency
of heat engines. The most fundamental concepts in thermodynamics are the system
and environment, and the most fundamental subjects are thermodynamic state and
process. A thermodynamic system is a macroscopic physical object. When a system
is in a thermodynamical equilibrium with an environment under certain conditions,
91
the proof that the two-body potential used by Lee and Yang does not meet the most
important interactions in physics such as the electromagnetic interaction [12].
As mentioned above, in most standard textbooks of statistical mechanics in the
definition of the partition function (2.4) no region for integration, or the supporting set
of the distribution function, is specified. When ergodicity is broken, as discussed in
the last section, the region in the phase space visited by a physical system changes,
and the supporting set of the distribution function changes correspondingly. The
determination of the supporting set depends on the physical problem under study.
For example, when dealing with a crystal, one has to consider firstly the symmetry
of the lattice, which then specifies the supporting set of the distribution function.
Gibbs introduced the grand canonical ensemble to deal with chemical reaction
systems. As an example, we may consider the reaction H + H → H 2 of the combination of two hydrogen atoms into a hydrogen molecule to compare quantum
mechanics with statistical mechanics. A treatment in quantum mechanics starts from
the Hamiltonian of two hydrogen atoms. The Born-Oppenheimer approximation is
used to separate out the degrees of freedom for nuclei, and the Schrödinger equation
of electrons is solved for electronic levels U i (R) as a function of the nuclear distance R. Bounded states are then interpreted as a hydrogen molecule. However, in
the treatment of statistical mechanics, one has to include both the hydrogen atoms
and the hydrogen molecules in the grand canonical ensemble at the beginning. From
a pure atomic Hamiltonian no hydrogen molecules would be derived from statistical mechanics. The phase space for a pure atomic system of hydrogen atoms is
different from that for a mixture of hydrogen atoms and molecules with regard to
the breaking of ergodicity. The first order phase transition may be regarded as the
simplest chemical reaction A → B, and thus should be treated using grand canonical
ensemble.
Whether in a simulation of molecular dynamics or in a Monte-Carlo simulation a
prerequisite is that no change is made in the supporting set of distribution function.
When applying accelerated or enhanced means for sampling, we must be careful.
This is often overlooked in the literature.
2.6 Analogue to Thermodynamics: Heat and Free Energy
Thermodynamic is a branch of physics studying the relation between heat and temperature and between energy and work. It defines macroscopic variables such as
internal energy, entropy and pressure, and describes universal relations among them
without regard to any specific property of specific matter. Such general rules are
represented by the four laws of thermodynamics.
The early motivation for thermodynamics was a desire to increase the efficiency
of heat engines. The most fundamental concepts in thermodynamics are the system
and environment, and the most fundamental subjects are thermodynamic state and
process. A thermodynamic system is a macroscopic physical object. When a system
is in a thermodynamical equilibrium with an environment under certain conditions,