1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
17
1.380649 10
−23 J/K , which marks the huge “distance” between the two. Then, once
the value of N A has been established and confirmed in many experiments, Eq. (1.39)
allows us to compute a macroscopic nonequilibrium property, the viscosity η, from an
equilibrium experiment in which no work is done: fluctuations are merely observed.
Note: viscosity does not exert any force when the fluid is at rest with respect to
its container or to the walls of an object in it. In a direct experimental measurement of the viscosity, the object in the fluid is dragged by a force F; then, viscosity
imposes a limit velocity v ∞ , and the mobility is defined by the linear response relation F = μv ∞ . Equation (1.39) bypasses all that work, and relates non-equilibrium
properties to equilibrium fluctuations, which are made visible by objects belonging
to an intermediate, mesoscopic, realm.
Hard to beat the beauty and importance of such a result, which settled once and for all
the dispute about the existence of atoms, whose mechanics provides a microscopic
description of the state of a macroscopic object. The theory of Brownian motion,
experimentally demonstrated by Perrin [9] dispelled all doubts, and several decades
later Feynman could state:
If, in some cataclysm, all of scientific knowledge had to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most
information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact,
or whatever you wish to call it) that all things are made of atoms—little particles that move
around in perpetual motion, attracting each other when they are a little distance apart, but
repelling upon being squeezed into one another. In that one sentence, you will see, there is
an enormous amount of information about the world, if just a little imagination and thinking
are applied.
Imagination is needed because even solving the corresponding equations of
motion, something so far impossible, would not help us understand the macroscopic
world. What would we do with myriads of graphs and numbers with very detailed
information on atoms trajectories, if we are concerned, for instance, about warming
up or cooling down something? These questions are much more naturally approached
by a very useful macroscopic description, not resting at all on microscopic theories:
Thermodynamics. However, there is catch: in order for the macroscopic approach to
be useful, its laws must be supplemented not only with a specification of the appropriate boundary conditions but with the values of thermophysical constants such as
the transport coefficients. These values cannot be predicted by the macroscopic theory, and must be supplied by experiments or derived from other approaches. One of
the goals of statistical mechanics is to predict these parameters from knowledge of
the interactions of the system’s constituent molecules.
1.3.1 Langevin Treatment
The solutions of the Langevin equation (1.15), or (1.21), define a stochastic process:
a process that is intrinsically probabilistic, in which only the statisical properties
of the variables of interest are relevant. There is an incredible variety of stochastic
17
1.380649 10
−23 J/K , which marks the huge “distance” between the two. Then, once
the value of N A has been established and confirmed in many experiments, Eq. (1.39)
allows us to compute a macroscopic nonequilibrium property, the viscosity η, from an
equilibrium experiment in which no work is done: fluctuations are merely observed.
Note: viscosity does not exert any force when the fluid is at rest with respect to
its container or to the walls of an object in it. In a direct experimental measurement of the viscosity, the object in the fluid is dragged by a force F; then, viscosity
imposes a limit velocity v ∞ , and the mobility is defined by the linear response relation F = μv ∞ . Equation (1.39) bypasses all that work, and relates non-equilibrium
properties to equilibrium fluctuations, which are made visible by objects belonging
to an intermediate, mesoscopic, realm.
Hard to beat the beauty and importance of such a result, which settled once and for all
the dispute about the existence of atoms, whose mechanics provides a microscopic
description of the state of a macroscopic object. The theory of Brownian motion,
experimentally demonstrated by Perrin [9] dispelled all doubts, and several decades
later Feynman could state:
If, in some cataclysm, all of scientific knowledge had to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most
information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact,
or whatever you wish to call it) that all things are made of atoms—little particles that move
around in perpetual motion, attracting each other when they are a little distance apart, but
repelling upon being squeezed into one another. In that one sentence, you will see, there is
an enormous amount of information about the world, if just a little imagination and thinking
are applied.
Imagination is needed because even solving the corresponding equations of
motion, something so far impossible, would not help us understand the macroscopic
world. What would we do with myriads of graphs and numbers with very detailed
information on atoms trajectories, if we are concerned, for instance, about warming
up or cooling down something? These questions are much more naturally approached
by a very useful macroscopic description, not resting at all on microscopic theories:
Thermodynamics. However, there is catch: in order for the macroscopic approach to
be useful, its laws must be supplemented not only with a specification of the appropriate boundary conditions but with the values of thermophysical constants such as
the transport coefficients. These values cannot be predicted by the macroscopic theory, and must be supplied by experiments or derived from other approaches. One of
the goals of statistical mechanics is to predict these parameters from knowledge of
the interactions of the system’s constituent molecules.
1.3.1 Langevin Treatment
The solutions of the Langevin equation (1.15), or (1.21), define a stochastic process:
a process that is intrinsically probabilistic, in which only the statisical properties
of the variables of interest are relevant. There is an incredible variety of stochastic