From mechanics to thermodynamics
W.D. Bauer
released
6.7.98 little additions 11.7.98 released
6.7.98 little additions 11.7.98
Abstract:
It is shown that the second law is obsolete in principle because it
can be derived as a consequence of the principle of least action if thetransition
from mechanics to thermodynamics is made. The second law corresponds to
the second variation of the least action functional and can be positive
or negative according to the physical problem. Therefrom it follows that
the possibility exists that the second law can reverse.
1. Introduction
Because moving particles are the deeper cause of heat, equilibrium
thermodynamics should be derivable from classical mechanics principally.In
this article we will present a derivation of equilibrium thermodynamics
from classical mechanics. It will be shown that the extremal principle
of the potentials follow from the principle of least action of classical
mechanics, and therefrom the second law becomes derivable as well for concrete
examples.
2. The Hamiltonian as minimizing function
It is a well known fact that the absolute value of the Lagrangian
in mechanics has to be minimized by a variational problem and that the
equations of motion can be derived from it as solution of this problem.In
this section we will prove that the same holds true for the Hamiltonian
if we restrict to the simple mechanic Hamiltonian
Therefore we formulate the
Proposition:
The Hamiltonian
is a minimized function of a variational problem which fulfills the equation
of motion as well.
Proof:
If we make the coordinate transformations 
, and if we redefine 
we can rewrite the Hamiltonian
We see from the second equation that -due to spatial invariance- the Hamiltonian can be looked at as a Lagrangian as well. It follows:
we
have a weak minimum for the Hamiltonian [1]. 
3. The transition from mechanics to thermodynamics
If the assumption is founded that the ergodic hypothesis can be applied
to a many particle system then the action integral mean total particle
energy in time is identical to the ensemble average or in mathematic language
with 
Proposition:
The conventional thermodynamic description can be found by a coordinate
transformation from the ensemble average of the Hamiltonian.
Proof: (for one dimensional systems)
From
It should be said that the total pressure 
and the total chemical potential [2] 
are terms which deviate from the conventional barometric measurable pressure
P and usual chemical potential 
because they contain the potential U 
additionally. The Shannon entropy derived above approaches the Boltzmann
entropy if it goes to a maximum [7,8]. If we extend Gibbs proof [3] regarding
the phase equilibrium in potential fields we see that these total quantities
and the temperature are constant over the whole volume of the system in
equilibrium, i.e. T =T=constant.This
fact allows us to deal with space dependent thermodynamic problems in potential
fields generally as hydrostatic or barometric pressure if we identify 
for example, comp. [2], or space dependent phase equilibria of mixtures
in fields.
4. The second law as consequence of the least action principle in mechanics
In the last section we showed that the mean Hamiltonian becomes
the inner energy in thermodynamics which becomes an extremum as a potential.
In this section we will show by an example that the second law follows
from the minimum principle of the thermodynamic potentials which itself
depends from the least action principle.
We imagine a cycle which includes a irreversible part of the closed
path at T=constant and V=constant, see fig.1a . The other part of the cycleis
reversible. Because the inner energy U is a potential it holds
Because 
due to minimizing of U during irreversible transitions it follows
and therefrom (because of 
)
which is the Clausius statement of the second law valid in this case.
It should be said that the analog considerations can be done for a
cycle with an irreversible part at P=constant and T=constant by using the
enthalpy H as the potential, see fig.1b.
Example:
In order to illustrate the general example given above we present here
as a concrete example a simplified version [4] of a thermodynamic cycle
which was proposed originally by Baltzar van Platen, the inventor of the
absorbtion refrigerator [5], see fig.2a-c. This purely isothermal cycle
with an periodic irreversibility proceeds in the lower region of the binary
P-x-phase diagram of a binary mixture 
.
Beginning at state 1 (point 1) at the dew line of the phase diagram, the
vapour is compressed reaching state 2 (point 2). In the course of this
compression vapour condenses to liquid. At state 2 the volume is split
into two parts (see points 2' and 2") by closing a tap, one volume part
2' contains only vapour and the other volume part 2" liquid and vapour.Then
both the volumes 2' and 2" are expanded separately up to the initial pressure
of the cycle and reach the states 3' and 3" in each compartments.Then the
tap between both the compartments is opened, and the initial state1 is
reached again on a irreversible path. Fig.2c represents the work area of
the van Platen cycle. More of this system and its material conditions in
[6].
According to our understanding of equilibrium thermodynamics the second
law is finally a consequence of the principle of least action. Contrary
to the conventional point of view it is not an axiom but only a consequence
of the mathematics of the problem. The situation is illustrated and summarized
in fig. 3 . Because the thermodynamic formalism can describe only the equilibrium,
the second law given addionally determines the direction of the process
if irreversible states occur.
As shown above this direction depends on the direction of the least
action integral of the corresponding many particle system. If the functional
has saddle point behavior then the second law in thermodynamics would have
to reverse in certain directions of the state space as well in order to
maintain consistency. Surprisingly this can happen really if if cyclic
varying fields are included into the consideration as we will show in the
next
web article[10]. We will discuss cases there where the second law reverses.
5.Conclusion
In this article we showed that the thermodynamic potential formalism
can be thougth to be stemming from the Hamilton many particle formalism.
It could be shown that the Hamiltonian is a potential which net amount
minimizes as well as the Lagrangian. In doing the ensemble average of many
particles the Hamiltonian becomes the inner energy, whose dynamical behavior
is summarized in the extremum principle of the potential which gives the
direction of the irreversible process but can not follow numerically the
processes in non-equilibrium. Therefrom, conventional and (as we
will see below) non-conventional
versions of the second law [10] follow dependent from the special problem
under consideration. Therefore it is easy understandable that many concepts
in thermodynamics have an analogon in mechanics. In Tab.1 the most important
features are summarized.
Tab.1: analogous features between thermodynamics and mechanics
| mechanics | thermodynamics |
| time mean or least action functional | ensemble average |
| Hamilton energy | inner energy |
| non-minimal state of functional | irreversible state |
| Legendre transformations, i.e. L, H | Legendre transformations, i.e. U, H, F, G |
| least action principle | extremum principle of potentials |
| second variation of functional | "second law" |
Acknowledgement: Thanks to Remi Cornwall for a constructive critique which helped to fill a hole in my argumentation.
Bibliography:
1) Bronstein-Semendjajew Taschenbuch der Mathematik, Harri Deutsch,Frankfurt,
1984
2) The term total potential stems from van der Waals and Kohnstam
see for example: V.Freise Chemische Thermodynamik BI Taschenbuch1973
3) J.M. Prausnitz, R.N. Lichtenthaler,E.G. de Azevedo
Molecular Thermodynamics of Fluid-Phase Equilibria, second edition,see
appendix.1
Prentice Hall, Englewood Cliffs, 1986
4) G. Hilscher Energie im Überfluß, Adolf Sponholtz, Hameln,1981
5) B. van Platen US patent 4.084.484 18.4.1978
6) W.D. Bauer, W. Muschik, Journal of Non-Equilibrium Thermodynamics(in
press)
probably vol.23, no.2 or 3, 1998
Second law induced existence conditions for isothermal 2-phase
region cyclic processes in binary mixtures
7) K. Huang, Statistical Mechanics, Wiley, New York, 1964
look under 4.1.: Boltzmann´s H-theorem
8) F. Topsoe, Information Theory, Teubner, Stuttgart, 1974
9) R.L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics I, Springer Berlin, 1992
10)see www.overunity-theory.de/bauer/index.html
fig.1a: isothermal reversible cycle with an irreversible path at V=constant,comp.
text
fig.1b: isothermal reversible cycle with an irreversible path at P=constant,comp. text
fig.2a: simplified van Platen cycle in a P-x-phase diagram of a binary mixture at constant temperature
fig.2c: total P-V diagram of a simplified van Platen cycle of a binary mixtures of constant temperature
fig.3 illustration of derivation of the second law from the principle of least action the direction of the second law is given by the second variation of the least action functional