1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
75
atoms is not suitable e.g. to determine whether someone has got the flu, not to mention
why one falls in love: a different perspective is surely needed for that.
Statisical mechanics in general makes use of concepts such as determinism and
randomness, probabilities and material properties, and it daringly mingles them. This
is necessary, given the breath of phenomena it intends to encompass. Understanding the foundation of statistical mechanics and, in particular, of its nonequilbrium
section, helps its tools to extend their applicability beyond the well explored traditional grounds.
Indeed, understanding what makes probability material, in some sense, and what
makes this identification fail in the cases in which the elementary constituents are
atoms and molecules, sheds light on the other uses of probability, even those in which
material properties are only vaguely present. These are most in need of investigation,
and are indeed investigated in probabilisitc terms, because no other tools seem to be
available or equally satisfactory. At the same time probability may be used, but with
little profit, if it is taken as an ethereal and mysterious entity.
Of course, statistical mechanics preserves all its interest in the study of the physical world, which is currently greatly developing, thanks to growing technological
abilities in handling mesoscopic and microscopic spatiotemporal scales, but also
in understanding geophysical scales, such as those concerning climate, oceans etc.
Last but not least, even astrophysical and cosmological research has recently been
approached from the nonequilbrium statistical mechanics point of view [47]. It will
not be surprising in the future to see more and more fields of research open their
doors to the nonequilbrium statistical mechanical approach.
Acknowledgements The author is indebted to Xiamen University and Huaqiao University for
unique and generous hospitality. Deep gratitude is in order to Prof. Hong Zhao and his group,
particularly Prof. Dahai He, for numerous deep and hearty scientific discussions, often accompanied
by great food. The author thanks Prof. Giovanni Ciccotti, for reading this manuscript, and for the
consequent intense, heated, exhilarating and enlightening discussions on foundational aspects of
thermodynamics and statistical physics, which helped me shape the present text.
Appendix 1: Exercise and Interpretation of Stochasticity
Consider a test particle of position x and velocity v, and a field particle of position
x 1 and velocity v 1 . Suppose they obey the following set of differential equations:
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
˙
x = v
˙
v = −γv + x 1
˙
x 1 = v 1
˙
v 1 = −kx 1
with initial conditions
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
x(0) = 0
v(0) = 0
x 1 (0) = x 10
v 1 (0) = −kx 10
(1.268)
where k = ˆ
k + 1 > 1 represents the effect of a spring and of the test particle on the
field particle, and we assume that the initial conditions for the test particle are known,
75
atoms is not suitable e.g. to determine whether someone has got the flu, not to mention
why one falls in love: a different perspective is surely needed for that.
Statisical mechanics in general makes use of concepts such as determinism and
randomness, probabilities and material properties, and it daringly mingles them. This
is necessary, given the breath of phenomena it intends to encompass. Understanding the foundation of statistical mechanics and, in particular, of its nonequilbrium
section, helps its tools to extend their applicability beyond the well explored traditional grounds.
Indeed, understanding what makes probability material, in some sense, and what
makes this identification fail in the cases in which the elementary constituents are
atoms and molecules, sheds light on the other uses of probability, even those in which
material properties are only vaguely present. These are most in need of investigation,
and are indeed investigated in probabilisitc terms, because no other tools seem to be
available or equally satisfactory. At the same time probability may be used, but with
little profit, if it is taken as an ethereal and mysterious entity.
Of course, statistical mechanics preserves all its interest in the study of the physical world, which is currently greatly developing, thanks to growing technological
abilities in handling mesoscopic and microscopic spatiotemporal scales, but also
in understanding geophysical scales, such as those concerning climate, oceans etc.
Last but not least, even astrophysical and cosmological research has recently been
approached from the nonequilbrium statistical mechanics point of view [47]. It will
not be surprising in the future to see more and more fields of research open their
doors to the nonequilbrium statistical mechanical approach.
Acknowledgements The author is indebted to Xiamen University and Huaqiao University for
unique and generous hospitality. Deep gratitude is in order to Prof. Hong Zhao and his group,
particularly Prof. Dahai He, for numerous deep and hearty scientific discussions, often accompanied
by great food. The author thanks Prof. Giovanni Ciccotti, for reading this manuscript, and for the
consequent intense, heated, exhilarating and enlightening discussions on foundational aspects of
thermodynamics and statistical physics, which helped me shape the present text.
Appendix 1: Exercise and Interpretation of Stochasticity
Consider a test particle of position x and velocity v, and a field particle of position
x 1 and velocity v 1 . Suppose they obey the following set of differential equations:
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
˙
x = v
˙
v = −γv + x 1
˙
x 1 = v 1
˙
v 1 = −kx 1
with initial conditions
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
x(0) = 0
v(0) = 0
x 1 (0) = x 10
v 1 (0) = −kx 10
(1.268)
where k = ˆ
k + 1 > 1 represents the effect of a spring and of the test particle on the
field particle, and we assume that the initial conditions for the test particle are known,