Second law article

From mechanics to thermodynamics
W.D. Bauer
released 6.7.98 little additions 11.7.98 some corrections 29.8.98

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It is shown that the second law is obsolete in principle because it can be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 the transition 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:

Because 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

the energy formula, W is the probability function and U is
the potential. In doing this we loose any time depedent information of the
system and we restrict ourselves to determine the total mean amount of the sum
of all energy values from the Hamiltonian. On the other hand the description of
the system becomes more simple. In principle we are making here the transition
from a many particle system ansatz to the conventional thermodynamic
description-as we will show in the following (not completely new [9])

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
follows
with the identifying definitions

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 . In our case we replace the isothermic path by an isentropic
which can be done without any problems. 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 isothermically 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]. The same process can be proceeded as well using benzene or another
highmolecular hydrocarcon because these materials have an T-S diagram phase
diagram which hangs over.

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.com/bauer/index.html

fig.1a: isentropic reversible cycle with an irreversible path at
V=constant,comp. text

fig.1b: isentropic 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.2b: simplified van Platen at constant temperature or
proceeded on reversible isentropes: 1-2 compression,2 splitting the volume, 2-3
separate expansion of both compartment, 3 opening the tap, 3-1 irreversible and
isothermal closing of the cycle

fig.2c: total P-V diagram of a simplified van Platen
cycle of a binary mixtures

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

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