7.3 Thermal Instabilities Connected to Phase Transformations 137
the transformation temperature per se, the probability is 0.5, as the particle may
be either in phase 1 or phase 2. Furthermore, an additional remark is necessary:
Until now, the discussion was related to one particle. Do things change if there is
instead of one particle an ensemble of many particles? Not at all! From statistical
thermodynamics it is well known that the probability to find one particle in a
certain phase or the concentration of particles belonging to an ensemble are
equivalent (the ergodic theorem).
Box 7.4 Some Terms Used in Thermodynamics
In the following, a few often-used terms are explained in a simplified way:
Ensemble
An ensemble consists of many identical objects; generally, the number of these
objects is known; however, this number may be infinite, too.
Isotherm
A process is isothermal if, for example, in the case of a phase transformation
or whatever else happens, the temperature does not change. Lastly, one assumes
that the process happens in a bath of infinite size and constant temperature.
Adiabatic
A process is adiabatic, if, for example, in the case of a phase transformation,
there is no flow of energy; therefore, the temperature changes. This is the idea
that there is an ideal thermal insulation around the objects (particles). Looking
at adiabatic ensembles, for example, undergoing phase transformations, one
has to distinguish two cases:
• Local enclosure: In this case each particle has an individual adiabatic enclosure [5].
• Global enclosure: The whole ensemble is in an adiabatic enclosure; therefore, heat exchange between the particles is possible [5].
Ergodic theorem
This is an important theorem in statistical physics, the “ergodic theorem”,
stemming from Boltzmann and Gibbs, saying, drastically simplified, that one
may replace a time average by an ensemble average. This sounds complicated;
however, it simplifies the considerations. Lastly, this theorem says that there
is no difference in the result if one takes one particle and repeats an experiment
say a thousand times or one takes a thousand particles, makes one experiment
and measures the property of each one of these particles. Hence, one may
interpret the experimental result either as the probability to find one particle
in a certain phase or as concentration, which is the number fraction, of particles
in this phase.
the transformation temperature per se, the probability is 0.5, as the particle may
be either in phase 1 or phase 2. Furthermore, an additional remark is necessary:
Until now, the discussion was related to one particle. Do things change if there is
instead of one particle an ensemble of many particles? Not at all! From statistical
thermodynamics it is well known that the probability to find one particle in a
certain phase or the concentration of particles belonging to an ensemble are
equivalent (the ergodic theorem).
Box 7.4 Some Terms Used in Thermodynamics
In the following, a few often-used terms are explained in a simplified way:
Ensemble
An ensemble consists of many identical objects; generally, the number of these
objects is known; however, this number may be infinite, too.
Isotherm
A process is isothermal if, for example, in the case of a phase transformation
or whatever else happens, the temperature does not change. Lastly, one assumes
that the process happens in a bath of infinite size and constant temperature.
Adiabatic
A process is adiabatic, if, for example, in the case of a phase transformation,
there is no flow of energy; therefore, the temperature changes. This is the idea
that there is an ideal thermal insulation around the objects (particles). Looking
at adiabatic ensembles, for example, undergoing phase transformations, one
has to distinguish two cases:
• Local enclosure: In this case each particle has an individual adiabatic enclosure [5].
• Global enclosure: The whole ensemble is in an adiabatic enclosure; therefore, heat exchange between the particles is possible [5].
Ergodic theorem
This is an important theorem in statistical physics, the “ergodic theorem”,
stemming from Boltzmann and Gibbs, saying, drastically simplified, that one
may replace a time average by an ensemble average. This sounds complicated;
however, it simplifies the considerations. Lastly, this theorem says that there
is no difference in the result if one takes one particle and repeats an experiment
say a thousand times or one takes a thousand particles, makes one experiment
and measures the property of each one of these particles. Hence, one may
interpret the experimental result either as the probability to find one particle
in a certain phase or as concentration, which is the number fraction, of particles
in this phase.