Chapter 2
On the Foundational Principles
of Statistical Mechanics
Wei-Mou Zheng
Abstract Several problems concerning the fundamental principles of statistical
mechanics, including the canonical ensemble theory, the supporting set of an ensemble distribution function, the correspondence to thermodynamics, irreversibility, and
time evolution, are briefly discussed.
Keywords Equilibrium statistical mechanics · Statistical ensemble · Entropy ·
Irreversibility
Here we first quote a few words from Chap. 7 of the book Twentieth Century Physics
by Domb [1]. “In 1902 Gibbs introduced the concept of an ensemble [2], a collection
of physical bodies following a statistical distribution . . . The properties of the body
are calculated by taking a statistical average over the ensemble, and fluctuations above
this average can be calculated using standard statistical procedures. . . . if the number
of constituents, N , of the body becomes large, the fluctuations will decrease, and the
averages can then be identified with thermodynamic properties in equilibrium. . . .
Gibbs showed that by identifying −kT ln Z N (where Z N is the partition function) with
the free energy of Helmholtz, F(T , V , N ), results were obtained which coincided
with thermodynamics.”
Gibbs introduced the grand canonical ensemble and chemical potentials, and used
the grand partition function (GPF) to deal with the thermodynamics of heterogeneous
substances. “Use of the GPF greatly simplifies the statistical mechanics of heterogeneous assemblies. . . . It took several decades for the scope and power of Gibbs
approach to be appreciated.”
Gibbs “constructed a general theory of statistical mechanics which came to be
seen as a foundation for all work in this field during the twentieth century. It is
therefore appropriate to recognize him as a great pioneer of modem physics.” “A
lucid series of lectures by Schrödinger [3] in Dublin in 1944 may well have been
responsible for putting Gibbs ideas . . . as the basis of statistical mechanics . . .”
W.-M. Zheng (B)
Institute of Theoretical Physics, Academia Sinica, 100190 Beijing, China
e-mail: zheng@itp.ac.cn
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
X.-Y. Liu (ed.), Frontiers and Progress of Current Soft Matter Research,
Soft and Biological Matter, https://doi.org/10.1007/978-981-15-9297-3_2
83
On the Foundational Principles
of Statistical Mechanics
Wei-Mou Zheng
Abstract Several problems concerning the fundamental principles of statistical
mechanics, including the canonical ensemble theory, the supporting set of an ensemble distribution function, the correspondence to thermodynamics, irreversibility, and
time evolution, are briefly discussed.
Keywords Equilibrium statistical mechanics · Statistical ensemble · Entropy ·
Irreversibility
Here we first quote a few words from Chap. 7 of the book Twentieth Century Physics
by Domb [1]. “In 1902 Gibbs introduced the concept of an ensemble [2], a collection
of physical bodies following a statistical distribution . . . The properties of the body
are calculated by taking a statistical average over the ensemble, and fluctuations above
this average can be calculated using standard statistical procedures. . . . if the number
of constituents, N , of the body becomes large, the fluctuations will decrease, and the
averages can then be identified with thermodynamic properties in equilibrium. . . .
Gibbs showed that by identifying −kT ln Z N (where Z N is the partition function) with
the free energy of Helmholtz, F(T , V , N ), results were obtained which coincided
with thermodynamics.”
Gibbs introduced the grand canonical ensemble and chemical potentials, and used
the grand partition function (GPF) to deal with the thermodynamics of heterogeneous
substances. “Use of the GPF greatly simplifies the statistical mechanics of heterogeneous assemblies. . . . It took several decades for the scope and power of Gibbs
approach to be appreciated.”
Gibbs “constructed a general theory of statistical mechanics which came to be
seen as a foundation for all work in this field during the twentieth century. It is
therefore appropriate to recognize him as a great pioneer of modem physics.” “A
lucid series of lectures by Schrödinger [3] in Dublin in 1944 may well have been
responsible for putting Gibbs ideas . . . as the basis of statistical mechanics . . .”
W.-M. Zheng (B)
Institute of Theoretical Physics, Academia Sinica, 100190 Beijing, China
e-mail: zheng@itp.ac.cn
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
X.-Y. Liu (ed.), Frontiers and Progress of Current Soft Matter Research,
Soft and Biological Matter, https://doi.org/10.1007/978-981-15-9297-3_2
83