ENTROPY AND THERMODYNAMIC PROBABILITY DI
ENTROPY AND THERMODYNAMIC PROBABILITY
DISTRIBUTION OVER PHASE SPACES
Ujjawal Krishnam1, Jason Kristiano2, Wounsuk Rhee3, Sridhar VR Prabhu4, Parth D.
Pandya5, Josephine Melia6, Kevin Limanta7
7
Massachusetts Institute Of Technology,77 Massachusetts Ave, Cambridge, MA 02139, USA
3
Seoul National University, 1 Gwanak-ro, Daehak-dong, Gwanak-gu, Seoul, South Korea
1,5
The Maharaja Sayajirao University of Baroda, University Road, Vadodara, Gujarat 390002
4University Of Cambridge, Emmanuel College, St Andrew's Street CB2 3AP Cambridgeshire UK
6
University of Illinois at Urbana-Champaign Champaign, IL USA
2
Universitas Indonesia, Kampus UI, Depok, Jawa Barat, Indonesia
March 14th; 2017
ABSTRACT
Thermodynamic probability(Ω) distribution over phase spaces is extensively studied. Entropy change(∆S) for
reversible motions in the single(μ-space positioning) and multiparticle system is coupled with Schrodinger Equation and
fractional fluctuations in thermodynamical ensemble for better approximations. Einstein’s theory on Brownian Motion is
treated with Γ-space microstate-macrostate correspondence on experimental relevance. Specifications of classically
defined non-meaningful arrangement of particles are supplemented with considerations of degenerate energy levels and
distinguishability factor among apparently indistinguishable particles. Further in paper, we investigate equivalence of
Sackur-Tetrode equation with classically obtained relation on thermodynamical probability. 3-D Markov Chain and Abel’s
summation are introduced over Sterling Approximation for calculation of most probable microstate in the accordance of
Bose-Einstein Distribution. Finally, we introduce abstract algebraic group structurisation on particles’ motion for abelian
and
non-abelian
characterisation
on
the
contours
of
phase
space
shifts.
Entropy and thermodynamic probability distribution over phase spaces
I.
INTRODUCTION
Statistical Physics aims at studying the parameters
on macroscopic scale with supplements of
consideration of microscopic properties, where
thermodynamics provides us with defined
macroscopic
measures
with
independent
parameters. However, the equation of state cannot
be obtained from the laws of thermodynamics,
experimental data play major role. Particularly
where the system consists of a large number of
particles, ordinary laws of mechanics could not be
used, as it is impossible to follow the motion of
each particle. Such difficulties can easily be solved
with statistical modeling, hence Statistical
Mechanics is taken in use to successfully solve
problems related to physical systems containing
large number of particles. Contrary to the
existence
of
advanced
statistics,
the
approximation and relevance at fundamental level
still exist as improperly defined concepts. In this
paper, we discuss the entropy for phase space
positions for particle systems. Further, we
reconsider
the
existing
concept
of
Thermodynamic Probability and define it with
various physical approaches in order to reduce
the
anomalies
and
to
obtain
better
approximations. Paper is organized as follows. In
Section II, we discuss our methodology. Section III
includes discussion on mathematical simulation
followed by conclusion.
compartment 2 of a vessel(ensemble) and its
return to initial μ-space δqaδpb, at this position, it is
quite important to look at microstate
configuration change of the particles(for a single
particle and multiparticle system). In our
probabilistic determination of entropy change, for
single particle system, we use Schrödinger
equation to estimate entropy change with
mathematical models, and hence we take
Boltzmann-Maxwellian
and
Clausius
interpretation(s) of Entropy to estimate the
entropy change for mutliparticle system.
1.1 Entropy Estimation for Single Particle
System
Fig.1: μ-space shift of particle
The number of microstates corresponding to specified
macrostate in a compartment is given by
Thermodynamic Probability, Ω. Schrödinger
equation can be written as,
Having
II.
METHODOLOGY
1. Entropy
in
positioning
where
reversible
μ-space
Considering an indistinguishable particle moving
from μ-space δqaδpb of compartment 1 to δqcδpd of
and
where
1
,
Hence,
and mathematical treatments lead
to,
.Hence,
Thus,
. On this line, we get the
proportional equivalence between Thermodynamic
probability and position-momenta corresponding to
μ-space, we get
Entropy and thermodynamic probability distribution over phase spaces
1
Similarly, considering positional shifts of phase
spaces, mathematical chain can be imagined as,
1
This results into S=0 for reversible μ-space shifts for
a single particle system.
1.2 Entropy Estimation for Multi-Particle
System
For a multi-particle system, we consider phase
space(i.e. μ-space) positioned particles in a
specified ensemble. In such multi-particle system,
when a particle changes phase space in the
ensemble it s not completely possible that the
particle will return to its initial phase space
ensuring that all particles(which have been
displaced from their initial positions in process)
will also return to their initial phase spaces. This
clearly indicates a certain minute(but not
completely negligible) change in microstate in
compartmental divisions of the ensemble.
Consequently position- momentum coordinates
will also be disturbed, hence phase space shifts
will occur. Further we consider the paths along
which particle can move to attain different phase
spaces. Considering here Clausius entropy
11 ), we find that entropy
concept(
∫
can be minimized to zero on reversibility with no
energy expenditure, if particle will prefer same
path under specified time interval providing
constraints remain defined. Contrarily, if particle
takes another route and in process spending more
energy(no matter it returns approximately to
initial phase space finally). Comparatively on large
scale, say on multi-particle ensemble level, when
large number of particles are considered under
defined isothermal and adiabatic conditions,
energy interchange cannot be ignored on such a
big scale. Hence, consideration of ensemble s
internal fluctuations is important. We can assume
S≈0(tending to Zero but not perfectly equal to
zero) also the entropy will tend to increase in such
cases.
While
1 represents the phase space shift
from
to
, the returning route may not be
same, hence on probabilistic determination, can
be interpreted as,
Hence δS
is possible under considerations of
certain conditions only.
1 . So,
As known,
1 1 where c is number of ensemble s
compartments and n indicates number of
distinguishable
particles.
As
∫
1
1
under normalization conditions where
f
defines
fractions
under
fluctuation
consideration, thermodynamic probability can be
expressed with provided fluctuations.
So,
(
√
)
1
Now with Clausius and Boltzmann-Maxwellian
interpretations on entropy, we estimate the
entropy difference for two random paths a and
b between two specified phase spaces. This
interprets
≈
1
with
Clausius interpretations On this line the perfect
nullity on entropy difference cannot be achieved,
hence nullifying factor is considerably important.
This yields,
Entropy and thermodynamic probability distribution over phase spaces
1
Nullifying factor, as defined, can be calculated with
fluctuation values which cannot be neglected. We
can easily correlated fluctuations with microstatemacrostate correspondence, so taking BoltzmannMaxwellian interpretations in consideration.
Thus,
Or,
[
(
√
)]
1
1
On this contour, we visualize the estimation of entropy
for single and multi-particle system.
2.
Thermodynamic
probability:
Finding
meanings of non-meaningful arrangements
Number of microstates corresponding to specific
macrostate is specified by thermodynamic
probability Ω, mathematically defined by
Combination as described in
1 considering
meaninful arrangement of particles classically On
is not considered which
this line, Permutation
may include so classically defined non-meaningful
arrangemnts, as such couting twice for two
particles’ mutually exchanged phase space positions
which Combination counts for once. Contrarily,
supposition of particles of same element with certain
classically suppressible distinguishabilities on their
microstate-macrostate correspondence count results
into certain anomalies. On a considerable scale, this
cannot
be
neglected,
consideration
of
distinguishability
among
apparently
indistinguishable
particles
becomes
quite
important[E.T. Jaynes], hence, in this section, we
estimate thermodynamic probability with better
approximations on statistical considerations.
2.1 Thermodynamic Probability Estimation on
Gamma Space specifications
we discuss a new way of determining the total
number of possible arrangements of same particles in
a container of volume . Here, we set the distribution
of particles as a continuous function of its
momentum, unlike the case in quantum physics
where states are discrete and quantized. Also, we will
assume a three-dimensional space, small particle
density, and no intermolecular interactions. In the
infinitesimal momentum range p p δp we will
assume that the degeneracy is
p , which is a
function of momentum, and the total number of
particles in that range is n p
. Here, we can notice
that the total number of particles, denoted by N
follows the relation N ∫ n p δp
Since we consider this situation in 3D container, we
can claim that p is proportional to p
, as we
can recall it from the expression d
r
. Also,
it is reasonable to assert that p is proportional to
the total volume
because larger volume allows
particles to be located at more diverse locations.
Thus, we can derive a simple relation as shown in
1 , where C is a constant.
p
1
For each interval p p δp , we can calculate the
possible number of arrangements, denoted by
p p δp , using the combination function, since
we assumed the particles to be identical.
pp
δp
≈
In
, we used the low-density assumption to
approximate the numerator as a power function. We
can further approximate this expression by using
Stirling’s approximation, demonstrated in
1,
to reach the value of ln p p δp .
lnN ≈ NlnN
ln
pp
N
Nln
δp ≈ n p δp ln
1
By using
, we obtain the expression for the
total number of possible arrangements
by
multiplying all
p p δp throughout the range
p
. We can calculate ln by integrating all of
Entropy and thermodynamic probability distribution over phase spaces
ln p p δp throughout the possible range, as we
follow in
ln
∫ ln
pp
δp
This way, we obtain the final expression for ln from
using the results discussed in
and
,
as follows.
ln
Nln eC
δp
ln
∫
proportional to N, and define
p
Nln eC
NlnN
N∫
is
, we can
simplify the expression obtained in
form of
in the
n p ln
From this expression, we notice the relevance of
Sackur-Tetrode equation with our approach, in a way
that the entropy or ln includes the term Nln
NlnN C N. This shows reasonable equivalence with
quantum mechanical expression under defined
restriction on n p .
2.2 Brownian Motion and Thermodynamic
Probability Estimation
Here we consider micro canonical ensemble, each
summing to form provided ensemble.
Let us define
(
with Γ-space considerations,
∫
1
)
As particles are in motion, E is solely kinetic,
hence calculations yield,
1
In addition to that, if we assume that n p
ln
of particle from one place to another in mean time, as
we get,
1
(
)
On the line, Energy as specified by
can be
taken further in mathematical grasp of
,
and hence, can obtain better approximation on
Thermodynamic Probability Estimation.
2.3
Thermodynamic Probability distribution
over Degenerate Energy Levels.
In this sub-section, we consider an energy level(j)
with degeneration gj and number of particles Nj.
On this outline, we can obtain a pattern of
arrangement of particles in specified energy
states, as illustrated as follows,
For gj=2 and Nj=2, we tabulate the arrangement
of particles in assumed energy states in Table 1,
State 1
State 2
a
b
b
a
ab
-
-
ba
Table 1: ωj=2
2
Similarly, we tabulate the energy state distribution
for gj=2 and Nj=2 in Table 2,
Or,
Or,
Taking experimentally verified Einstein’s theory of
Brownian Motion, we can calculate the displacement
Entropy and thermodynamic probability distribution over phase spaces
State 1
State 2
State 3
a
b
-
b
-
b
b
a
-
b
-
a
-
a
b
-
b
a
ab
-
-
-
ab
-
-
-
ab
Table 2: ωj=3
2
Hence, we observe a pattern on energy state
distribution Table 1 and Table 2, and that brilliantly
interprets
1 .
Succeeding this,
Thermodynamic Probability can be described as,
[∏
}
]{
∏
Where, underlined part of R.H.S. in
is
distinguishablity factor specifically defined on
particles
with
energy
level
interchange
correspondence.
ᴨ
as introduced in underlined part,
if defined on j=1, reduces
{
3.
∏
Bose-Einstein
approximations
to,
}
Distribution
(
1)
1)
4. Abstract Algebraic implications on
particle movement
Here we define G as a system of particles(i.e.
ensemble) . Consideration of ‘n’ number of particles
in the vessel as a finite Group and two compartments’
specification are supplemented with an assumption of
n/2 number of particles’ distribution in each
compartment.
Then we have,
and
BE statistics defines Thermodynamic Probability as
follows,
(
Most probable microstate corresponds to the state of
maximum thermodynamic probability and results into
Bose-Einstein’s distribution law for a class of
particles among various energy levels for a system
obeying Bose-Einstein Statistics. The approximation
technique used in process plugged with Lagrangian
method of undetermined multipliers is Sterling
approximation, and in case used for very large
number of particles. Oppositely, if condition includes
restricted distribution of selective particles, sterling
approximation may not prove to be fruitful. Under
this condition, the influential spheres of particles can
be considered with Abel’s summation and Markov
chain applications in three dimensions; and under
specified pattern, the bad approximation can be
reduced. Despite our efforts to define good
approximation, it is still difficult to consider the
good outline and boundary conditions of
approximations, hence in next section, we
investigate the distribution of particles in defined
ensemble with pure implications of group
structures of abstract algebra.
Where, denotes the number of ways in which ni
particles are to be distributed in gi cells in the ith
compartment.
{
compartment,
{
And,
}
for
one
}
for
second compartment, for even number of particles.
Similarly for n odd number of particles, group can be
defined as,
{
}
for
one compartment,
And,
{
Entropy and thermodynamic probability distribution over phase spaces
}
for
second compartment. In case of odd number of
particles, we cannot consider that both compartments
have same number of particles.
It is quite important to note here that negative sign is
for representational purpose and to indicate only the
particles in opposite compartment taken one in
proximity.
Hence, we define an operator ‘ʘ’ on system
suggesting the movements of particles. So, we can
define Group (G, ʘ) for the ensemble.
Closure Property: If we take two particles a1 and
a2, as they are moving, their movements will
always be defined on G as a1&a2∊G
Associative Property: Now if we take three
particles a1, a2, a3. As they are in motion, either we
focus on the movements on the lines of phase
space shifts of a2 and a3 excluding a1 for
considerable point of time. Or we consider other
two particles excluding one among three particles
from the system of n number of particles with
provided thermodynamic variables remain
defined.
a a
∊
Existence of Identity: On the line, we can imagine
the existence of a particle a0∊G such that phase
space shifts of such a particle will not affect the
movement of other particle.
∊
Existence of Inverse: Mutual phase space shifts of
particles from one compartment to another
without affecting the ensemble provides us with
an insight on existence of inverse.
∊
1
For indistinguishable particles, the commutativity
property holds good defining abelian groups of
particles. Contrarily, the correspondence of
particles generalized positions and momenta with
non-abelian property defines permutation groups,
and hence, can be used for simulated studies of
position-momenta shifts with algebraic analogues.
III.
RESULTS AND DISCUSSION
Entropy and Thermodynamic probability are
reconsidered with mathematical simulations. For
a single particle system, or for considerable single
particle motion in the ensemble, we find S=0 as
1 tells so. For the multiparticle system, the
entropy change is significantly evident on small
scale but not totally negligible as we find through
mathematical treatments from
1 to
1 .
On this line, we introduce nullifying factor taking
thermodynamic probabilistic fluctuations on
count.
Further, thermodynamic probability is estimated
with various measures. In Section 2.1, we
estimate thermodynamic probability with 3-D
considerations of particles phase shifts.
correlates quantum statistical relation with
classical analogues and hence, provides a good
insight on thermodynamic probability estimation.
With
to
,
we
define
thermodynamic probability on the lines of
Einstein s
theory
on
Brownian
Motion.
Thermodynamic probability is defined on the
basis of degenerate energy levels in Section 2.3.
Distinguishability factor is assumed with energy
level interchange correspondence. These suggest
new approaches to understand thermodynamic
probability. Later, we investigate approximation
techniques used in Bose-Einstein statistics and
suggest statistical markups on markov chain
consideration with abel s summation methods to
estimate most probable microstate with better
approximation implications. Algebraic group
structurisation of particles in the ensemble is
introduced for abelian and non abelian properties,
and we consider new mathematical models to
Entropy and thermodynamic probability distribution over phase spaces
understand ensemble conditions for particles
motion.
IV.
CONCLUSION
Our studies suggest new methods for
thermodynamic probability estimation. Entropy
change for a single and multiparticle systems of
indistinguishable particles under specified
reversible condition is studied and it suggests that
fluctuations are important while nullifying
entropy change. Approximation method, used in
Bose-Einstein Statistics, as Sterling approximation
does not seem effective on selective number of
particles, hence we introduce Abel s summation
and 3-D markov chain outline with influential
spheres consideration. Finally, we put forth the
implications of abstract algebraic structurisation
on particles phase space shifts and consider them
under abelian and non-abelian properties, thus,
can be useful for understanding thermodynamic
probability and phase space shifts.
V.
ACKNOWLEDGMENTS
Authors would like to thank Professor Dr Prafulla K.
Jha, Department Of Physics, The M.S. University Of
Baroda, Professor K. Muralidharan, Department Of
Statistics, The M.S. University Of Baroda, Professor
Dr Abhas Mitra, HBNI Mumbai, Dr Himadri
Barman, IMSc Chennai, Dr Geetanjali Sethi, St.
Stephens College, University Of Delhi, for their
valuable suggestions and guidances. Authors would
like to thank Department Of Physics, Faculty Of
Science, The M.S. University Of Baroda and
Department Of Science and Technology, Government
Of India for academic assistance and project
facilitation.
One
of
Authors(U.K.)
thanks
Academia.edu for editorial privileges and hence,
facilitating
the project work. Resources made
available at Smt. Hansa Mehta Library are highly
appreciated.
VI.
REFERENCES
[1] Bose
(1924).
"Plancks
Gesetz
und
Lichtquantenhypothese",
Zeitschrift
für
Physik 26:178–181
[2] Jaynes, E.T.(1965), American Journal Of
Physics, 33, 391-8
[3] Ludwig
Boltzmann(1896
and
1898).
Vorlesungen uber Gastheorie. J.A. Barth,
Leipzig.
[4] Ludwig Boltzmann (1866), Über die
Mechanische
Bedeutung
des
Zweiten
Hauptsatzes der Wärmetheorie". Wiener
Berichte. 53: 195–220.
[5] Curtright, T. L.; Zachos, C. K. (2012).
"Quantum Mechanics in Phase Space". Asia
Pacific Physics Newsletter. 01: 37
[6] A. Einstein, Investigations of the Theory of
Brownian Movement (Dover, 1956)
[7] Nernst, Walther (1888). "Zur Kinetik der in
Lösung befindlichen Körper". Z. Phys. Chem.
9: 613–637
[8] Clausius, Rudolf (1850). On the Motive Power
of Heat, and on the Laws which can be
deduced from it for the Theory of Heat.
Poggendorff's Annalen der Physick, LXXIX
(Dover Reprint). ISBN 0-486-59065-8
[9] Clausius, Rudolf (1865). Ueber verschiedene
für die Anwendung bequeme Formen der
Hauptgleichungen
der
mechanischen
Wärmetheorie:
vorgetragen
in
der
naturforsch. Gesellschaft den 24. April 1865.
p. 46
[10]
Hillar, Christopher; Rhea, Darren (2007).
"Automorphisms of finite abelian groups".
American Mathematical Monthly. 114 (10):
917–923
[11]
Cameron, Peter J. (1994), Combinatorics:
Topics, Techniques, Algorithms, Cambridge
University Press, ISBN 0-521-45761-0
Entropy and thermodynamic probability distribution over phase spaces
DISTRIBUTION OVER PHASE SPACES
Ujjawal Krishnam1, Jason Kristiano2, Wounsuk Rhee3, Sridhar VR Prabhu4, Parth D.
Pandya5, Josephine Melia6, Kevin Limanta7
7
Massachusetts Institute Of Technology,77 Massachusetts Ave, Cambridge, MA 02139, USA
3
Seoul National University, 1 Gwanak-ro, Daehak-dong, Gwanak-gu, Seoul, South Korea
1,5
The Maharaja Sayajirao University of Baroda, University Road, Vadodara, Gujarat 390002
4University Of Cambridge, Emmanuel College, St Andrew's Street CB2 3AP Cambridgeshire UK
6
University of Illinois at Urbana-Champaign Champaign, IL USA
2
Universitas Indonesia, Kampus UI, Depok, Jawa Barat, Indonesia
March 14th; 2017
ABSTRACT
Thermodynamic probability(Ω) distribution over phase spaces is extensively studied. Entropy change(∆S) for
reversible motions in the single(μ-space positioning) and multiparticle system is coupled with Schrodinger Equation and
fractional fluctuations in thermodynamical ensemble for better approximations. Einstein’s theory on Brownian Motion is
treated with Γ-space microstate-macrostate correspondence on experimental relevance. Specifications of classically
defined non-meaningful arrangement of particles are supplemented with considerations of degenerate energy levels and
distinguishability factor among apparently indistinguishable particles. Further in paper, we investigate equivalence of
Sackur-Tetrode equation with classically obtained relation on thermodynamical probability. 3-D Markov Chain and Abel’s
summation are introduced over Sterling Approximation for calculation of most probable microstate in the accordance of
Bose-Einstein Distribution. Finally, we introduce abstract algebraic group structurisation on particles’ motion for abelian
and
non-abelian
characterisation
on
the
contours
of
phase
space
shifts.
Entropy and thermodynamic probability distribution over phase spaces
I.
INTRODUCTION
Statistical Physics aims at studying the parameters
on macroscopic scale with supplements of
consideration of microscopic properties, where
thermodynamics provides us with defined
macroscopic
measures
with
independent
parameters. However, the equation of state cannot
be obtained from the laws of thermodynamics,
experimental data play major role. Particularly
where the system consists of a large number of
particles, ordinary laws of mechanics could not be
used, as it is impossible to follow the motion of
each particle. Such difficulties can easily be solved
with statistical modeling, hence Statistical
Mechanics is taken in use to successfully solve
problems related to physical systems containing
large number of particles. Contrary to the
existence
of
advanced
statistics,
the
approximation and relevance at fundamental level
still exist as improperly defined concepts. In this
paper, we discuss the entropy for phase space
positions for particle systems. Further, we
reconsider
the
existing
concept
of
Thermodynamic Probability and define it with
various physical approaches in order to reduce
the
anomalies
and
to
obtain
better
approximations. Paper is organized as follows. In
Section II, we discuss our methodology. Section III
includes discussion on mathematical simulation
followed by conclusion.
compartment 2 of a vessel(ensemble) and its
return to initial μ-space δqaδpb, at this position, it is
quite important to look at microstate
configuration change of the particles(for a single
particle and multiparticle system). In our
probabilistic determination of entropy change, for
single particle system, we use Schrödinger
equation to estimate entropy change with
mathematical models, and hence we take
Boltzmann-Maxwellian
and
Clausius
interpretation(s) of Entropy to estimate the
entropy change for mutliparticle system.
1.1 Entropy Estimation for Single Particle
System
Fig.1: μ-space shift of particle
The number of microstates corresponding to specified
macrostate in a compartment is given by
Thermodynamic Probability, Ω. Schrödinger
equation can be written as,
Having
II.
METHODOLOGY
1. Entropy
in
positioning
where
reversible
μ-space
Considering an indistinguishable particle moving
from μ-space δqaδpb of compartment 1 to δqcδpd of
and
where
1
,
Hence,
and mathematical treatments lead
to,
.Hence,
Thus,
. On this line, we get the
proportional equivalence between Thermodynamic
probability and position-momenta corresponding to
μ-space, we get
Entropy and thermodynamic probability distribution over phase spaces
1
Similarly, considering positional shifts of phase
spaces, mathematical chain can be imagined as,
1
This results into S=0 for reversible μ-space shifts for
a single particle system.
1.2 Entropy Estimation for Multi-Particle
System
For a multi-particle system, we consider phase
space(i.e. μ-space) positioned particles in a
specified ensemble. In such multi-particle system,
when a particle changes phase space in the
ensemble it s not completely possible that the
particle will return to its initial phase space
ensuring that all particles(which have been
displaced from their initial positions in process)
will also return to their initial phase spaces. This
clearly indicates a certain minute(but not
completely negligible) change in microstate in
compartmental divisions of the ensemble.
Consequently position- momentum coordinates
will also be disturbed, hence phase space shifts
will occur. Further we consider the paths along
which particle can move to attain different phase
spaces. Considering here Clausius entropy
11 ), we find that entropy
concept(
∫
can be minimized to zero on reversibility with no
energy expenditure, if particle will prefer same
path under specified time interval providing
constraints remain defined. Contrarily, if particle
takes another route and in process spending more
energy(no matter it returns approximately to
initial phase space finally). Comparatively on large
scale, say on multi-particle ensemble level, when
large number of particles are considered under
defined isothermal and adiabatic conditions,
energy interchange cannot be ignored on such a
big scale. Hence, consideration of ensemble s
internal fluctuations is important. We can assume
S≈0(tending to Zero but not perfectly equal to
zero) also the entropy will tend to increase in such
cases.
While
1 represents the phase space shift
from
to
, the returning route may not be
same, hence on probabilistic determination, can
be interpreted as,
Hence δS
is possible under considerations of
certain conditions only.
1 . So,
As known,
1 1 where c is number of ensemble s
compartments and n indicates number of
distinguishable
particles.
As
∫
1
1
under normalization conditions where
f
defines
fractions
under
fluctuation
consideration, thermodynamic probability can be
expressed with provided fluctuations.
So,
(
√
)
1
Now with Clausius and Boltzmann-Maxwellian
interpretations on entropy, we estimate the
entropy difference for two random paths a and
b between two specified phase spaces. This
interprets
≈
1
with
Clausius interpretations On this line the perfect
nullity on entropy difference cannot be achieved,
hence nullifying factor is considerably important.
This yields,
Entropy and thermodynamic probability distribution over phase spaces
1
Nullifying factor, as defined, can be calculated with
fluctuation values which cannot be neglected. We
can easily correlated fluctuations with microstatemacrostate correspondence, so taking BoltzmannMaxwellian interpretations in consideration.
Thus,
Or,
[
(
√
)]
1
1
On this contour, we visualize the estimation of entropy
for single and multi-particle system.
2.
Thermodynamic
probability:
Finding
meanings of non-meaningful arrangements
Number of microstates corresponding to specific
macrostate is specified by thermodynamic
probability Ω, mathematically defined by
Combination as described in
1 considering
meaninful arrangement of particles classically On
is not considered which
this line, Permutation
may include so classically defined non-meaningful
arrangemnts, as such couting twice for two
particles’ mutually exchanged phase space positions
which Combination counts for once. Contrarily,
supposition of particles of same element with certain
classically suppressible distinguishabilities on their
microstate-macrostate correspondence count results
into certain anomalies. On a considerable scale, this
cannot
be
neglected,
consideration
of
distinguishability
among
apparently
indistinguishable
particles
becomes
quite
important[E.T. Jaynes], hence, in this section, we
estimate thermodynamic probability with better
approximations on statistical considerations.
2.1 Thermodynamic Probability Estimation on
Gamma Space specifications
we discuss a new way of determining the total
number of possible arrangements of same particles in
a container of volume . Here, we set the distribution
of particles as a continuous function of its
momentum, unlike the case in quantum physics
where states are discrete and quantized. Also, we will
assume a three-dimensional space, small particle
density, and no intermolecular interactions. In the
infinitesimal momentum range p p δp we will
assume that the degeneracy is
p , which is a
function of momentum, and the total number of
particles in that range is n p
. Here, we can notice
that the total number of particles, denoted by N
follows the relation N ∫ n p δp
Since we consider this situation in 3D container, we
can claim that p is proportional to p
, as we
can recall it from the expression d
r
. Also,
it is reasonable to assert that p is proportional to
the total volume
because larger volume allows
particles to be located at more diverse locations.
Thus, we can derive a simple relation as shown in
1 , where C is a constant.
p
1
For each interval p p δp , we can calculate the
possible number of arrangements, denoted by
p p δp , using the combination function, since
we assumed the particles to be identical.
pp
δp
≈
In
, we used the low-density assumption to
approximate the numerator as a power function. We
can further approximate this expression by using
Stirling’s approximation, demonstrated in
1,
to reach the value of ln p p δp .
lnN ≈ NlnN
ln
pp
N
Nln
δp ≈ n p δp ln
1
By using
, we obtain the expression for the
total number of possible arrangements
by
multiplying all
p p δp throughout the range
p
. We can calculate ln by integrating all of
Entropy and thermodynamic probability distribution over phase spaces
ln p p δp throughout the possible range, as we
follow in
ln
∫ ln
pp
δp
This way, we obtain the final expression for ln from
using the results discussed in
and
,
as follows.
ln
Nln eC
δp
ln
∫
proportional to N, and define
p
Nln eC
NlnN
N∫
is
, we can
simplify the expression obtained in
form of
in the
n p ln
From this expression, we notice the relevance of
Sackur-Tetrode equation with our approach, in a way
that the entropy or ln includes the term Nln
NlnN C N. This shows reasonable equivalence with
quantum mechanical expression under defined
restriction on n p .
2.2 Brownian Motion and Thermodynamic
Probability Estimation
Here we consider micro canonical ensemble, each
summing to form provided ensemble.
Let us define
(
with Γ-space considerations,
∫
1
)
As particles are in motion, E is solely kinetic,
hence calculations yield,
1
In addition to that, if we assume that n p
ln
of particle from one place to another in mean time, as
we get,
1
(
)
On the line, Energy as specified by
can be
taken further in mathematical grasp of
,
and hence, can obtain better approximation on
Thermodynamic Probability Estimation.
2.3
Thermodynamic Probability distribution
over Degenerate Energy Levels.
In this sub-section, we consider an energy level(j)
with degeneration gj and number of particles Nj.
On this outline, we can obtain a pattern of
arrangement of particles in specified energy
states, as illustrated as follows,
For gj=2 and Nj=2, we tabulate the arrangement
of particles in assumed energy states in Table 1,
State 1
State 2
a
b
b
a
ab
-
-
ba
Table 1: ωj=2
2
Similarly, we tabulate the energy state distribution
for gj=2 and Nj=2 in Table 2,
Or,
Or,
Taking experimentally verified Einstein’s theory of
Brownian Motion, we can calculate the displacement
Entropy and thermodynamic probability distribution over phase spaces
State 1
State 2
State 3
a
b
-
b
-
b
b
a
-
b
-
a
-
a
b
-
b
a
ab
-
-
-
ab
-
-
-
ab
Table 2: ωj=3
2
Hence, we observe a pattern on energy state
distribution Table 1 and Table 2, and that brilliantly
interprets
1 .
Succeeding this,
Thermodynamic Probability can be described as,
[∏
}
]{
∏
Where, underlined part of R.H.S. in
is
distinguishablity factor specifically defined on
particles
with
energy
level
interchange
correspondence.
ᴨ
as introduced in underlined part,
if defined on j=1, reduces
{
3.
∏
Bose-Einstein
approximations
to,
}
Distribution
(
1)
1)
4. Abstract Algebraic implications on
particle movement
Here we define G as a system of particles(i.e.
ensemble) . Consideration of ‘n’ number of particles
in the vessel as a finite Group and two compartments’
specification are supplemented with an assumption of
n/2 number of particles’ distribution in each
compartment.
Then we have,
and
BE statistics defines Thermodynamic Probability as
follows,
(
Most probable microstate corresponds to the state of
maximum thermodynamic probability and results into
Bose-Einstein’s distribution law for a class of
particles among various energy levels for a system
obeying Bose-Einstein Statistics. The approximation
technique used in process plugged with Lagrangian
method of undetermined multipliers is Sterling
approximation, and in case used for very large
number of particles. Oppositely, if condition includes
restricted distribution of selective particles, sterling
approximation may not prove to be fruitful. Under
this condition, the influential spheres of particles can
be considered with Abel’s summation and Markov
chain applications in three dimensions; and under
specified pattern, the bad approximation can be
reduced. Despite our efforts to define good
approximation, it is still difficult to consider the
good outline and boundary conditions of
approximations, hence in next section, we
investigate the distribution of particles in defined
ensemble with pure implications of group
structures of abstract algebra.
Where, denotes the number of ways in which ni
particles are to be distributed in gi cells in the ith
compartment.
{
compartment,
{
And,
}
for
one
}
for
second compartment, for even number of particles.
Similarly for n odd number of particles, group can be
defined as,
{
}
for
one compartment,
And,
{
Entropy and thermodynamic probability distribution over phase spaces
}
for
second compartment. In case of odd number of
particles, we cannot consider that both compartments
have same number of particles.
It is quite important to note here that negative sign is
for representational purpose and to indicate only the
particles in opposite compartment taken one in
proximity.
Hence, we define an operator ‘ʘ’ on system
suggesting the movements of particles. So, we can
define Group (G, ʘ) for the ensemble.
Closure Property: If we take two particles a1 and
a2, as they are moving, their movements will
always be defined on G as a1&a2∊G
Associative Property: Now if we take three
particles a1, a2, a3. As they are in motion, either we
focus on the movements on the lines of phase
space shifts of a2 and a3 excluding a1 for
considerable point of time. Or we consider other
two particles excluding one among three particles
from the system of n number of particles with
provided thermodynamic variables remain
defined.
a a
∊
Existence of Identity: On the line, we can imagine
the existence of a particle a0∊G such that phase
space shifts of such a particle will not affect the
movement of other particle.
∊
Existence of Inverse: Mutual phase space shifts of
particles from one compartment to another
without affecting the ensemble provides us with
an insight on existence of inverse.
∊
1
For indistinguishable particles, the commutativity
property holds good defining abelian groups of
particles. Contrarily, the correspondence of
particles generalized positions and momenta with
non-abelian property defines permutation groups,
and hence, can be used for simulated studies of
position-momenta shifts with algebraic analogues.
III.
RESULTS AND DISCUSSION
Entropy and Thermodynamic probability are
reconsidered with mathematical simulations. For
a single particle system, or for considerable single
particle motion in the ensemble, we find S=0 as
1 tells so. For the multiparticle system, the
entropy change is significantly evident on small
scale but not totally negligible as we find through
mathematical treatments from
1 to
1 .
On this line, we introduce nullifying factor taking
thermodynamic probabilistic fluctuations on
count.
Further, thermodynamic probability is estimated
with various measures. In Section 2.1, we
estimate thermodynamic probability with 3-D
considerations of particles phase shifts.
correlates quantum statistical relation with
classical analogues and hence, provides a good
insight on thermodynamic probability estimation.
With
to
,
we
define
thermodynamic probability on the lines of
Einstein s
theory
on
Brownian
Motion.
Thermodynamic probability is defined on the
basis of degenerate energy levels in Section 2.3.
Distinguishability factor is assumed with energy
level interchange correspondence. These suggest
new approaches to understand thermodynamic
probability. Later, we investigate approximation
techniques used in Bose-Einstein statistics and
suggest statistical markups on markov chain
consideration with abel s summation methods to
estimate most probable microstate with better
approximation implications. Algebraic group
structurisation of particles in the ensemble is
introduced for abelian and non abelian properties,
and we consider new mathematical models to
Entropy and thermodynamic probability distribution over phase spaces
understand ensemble conditions for particles
motion.
IV.
CONCLUSION
Our studies suggest new methods for
thermodynamic probability estimation. Entropy
change for a single and multiparticle systems of
indistinguishable particles under specified
reversible condition is studied and it suggests that
fluctuations are important while nullifying
entropy change. Approximation method, used in
Bose-Einstein Statistics, as Sterling approximation
does not seem effective on selective number of
particles, hence we introduce Abel s summation
and 3-D markov chain outline with influential
spheres consideration. Finally, we put forth the
implications of abstract algebraic structurisation
on particles phase space shifts and consider them
under abelian and non-abelian properties, thus,
can be useful for understanding thermodynamic
probability and phase space shifts.
V.
ACKNOWLEDGMENTS
Authors would like to thank Professor Dr Prafulla K.
Jha, Department Of Physics, The M.S. University Of
Baroda, Professor K. Muralidharan, Department Of
Statistics, The M.S. University Of Baroda, Professor
Dr Abhas Mitra, HBNI Mumbai, Dr Himadri
Barman, IMSc Chennai, Dr Geetanjali Sethi, St.
Stephens College, University Of Delhi, for their
valuable suggestions and guidances. Authors would
like to thank Department Of Physics, Faculty Of
Science, The M.S. University Of Baroda and
Department Of Science and Technology, Government
Of India for academic assistance and project
facilitation.
One
of
Authors(U.K.)
thanks
Academia.edu for editorial privileges and hence,
facilitating
the project work. Resources made
available at Smt. Hansa Mehta Library are highly
appreciated.
VI.
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Entropy and thermodynamic probability distribution over phase spaces