#  On Polymer Statistical Mechanics: From Gaussian Distribution to Maxwell-
Boltzmann Distribution to Fermi-Dirac Distribution

Lixiang Yang  Department of Mechanical and Aerospace Engineering, The Ohio
State University, Columbus OH, 43210 Email<EMAIL_ADDRESS>
###### Abstract

Macroscopic mechanical properties of polymers are determined by their
microscopic molecular chain distribution. Due to randomness of these molecular
chains, probability theory has been used to find their micro-states and energy
distribution. In this paper, aided by central limit theorem and mixed Bayes
rule, we showed that entropy elasticity based on Gaussian distribution is
questionable. By releasing freely jointed chain assumption, we found that
there is energy redistribution when each bond of a molecular chain changes its
length. Therefore, we have to change Gaussian distribution used in polymer
elasticity to Maxwell-Boltzmann distribution. Since Maxwell-Boltzmann
distribution is only a good energy description for gas molecules, we found a
mathematical path to change Maxwell-Boltzmann distribution to Fermi-Dirac
distribution based on molecular chain structures. Because a molecular chain
can be viewed as many monomers glued by covalent electrons, Fermi-Dirac
distribution describes the probability of covalent electron occupancy in
micro-states for solids such as polymers. Mathematical form of Fermi-Dirac
distribution is logistic function. Mathematical simplicity of Fermi-Dirac
distribution makes many hard mechanics problems easy to understand.
Generalized logistic function or Fermi-Dirac distribution function was able to
understand many polymer mechanics problems such as viscoelasticity [1],
viscoplasticity [2], shear band and necking [3], and ultrasonic bonding [4].

Keywords: Gaussian Statistical Mechanics; Polymer Physics; Mixed Bayes Rule;
Conditional Probability Density Functions; Maxwell-Boltzmann Distribution;
Fermi-Dirac Distribution

##  1 Introduction to Entropy Elasticity

Mechanical behaviors of rubbers are largely determined by its micro-structures
[5, 6, 7, 8]. Since rubbers are microscopically made of randomly distributed
molecular chains, probability theory is introduced to understand their stress-
strain relationship [9]. When rubbers are stretched under external forces,
molecular chains will get stretched and change their configurations. After a
molecular chain changes from one configuration to another one, its entropy
will change too.

In entropy rubber elasticity, entropy instead of internal energy is considered
as the driving force of mechanical response [10]. So force and displacement or
stress-strain relationship is derived by relationship between probability
distribution of molecular chains and entropy of the system [11].

For a single chain, with one end fixed, the probability to find the other end
of the chain in a small box will obey Gaussian distribution or normal
distribution. Flory[12] derived Gaussian distribution for a random freely
jointed chain in one dimension. The distance of a molecular chain from one end
to the other end is projected to x-axis which can be treated as a random
variable $S_{n}$. The probability distribution of $S_{n}$ can be considered as
the corresponding excess of heads in a series of coin tosses. In another word,
random variable $S_{n}$ is a binomial random variable with probability of each
trial 0.5. In chapter X appendix A [12], Flory also showed the random variable
$S_{n}$ will become normal distribution if let the number of the bond vectors
of a molecular chain go to infinity. One dimensional Gaussian distribution
density function of end to end distance of a molecular chain in $x$ direction
was given as

$G(x,N)=(\,\frac{3}{2\pi Nb^{2}})\,^{1/2}\mathbf{e}^{-3x^{2}/2Nb^{2}},$ (1.1)

where $N$ is the number of Kuhn segments of a molecular chain and $b$ is given
as Kuhn length. By assuming Gaussian distributions in $x$, $y$ and $z$
directions are independent of each other, one dimensional Gaussian
distribution can be extended to become three dimensional Gaussian distribution
[13]

$G(\mathbf{r},N)=(\,\frac{3}{2\pi
Nb^{2}})\,^{3/2}\mathbf{e}^{-3r^{2}/2Nb^{2}},$ (1.2)

where $r$ is the magnitude of the chain displacement vector $\mathbf{r}$, i.e.

$r^{2}=x^{2}+y^{2}+z^{2}.$

In fact, what Flory derived can be mathematically proved by using central
limit theorem. Based on central limit theorem, the number of segment of each
molecular chain is not necessarily going to infinity to be a Gaussian
distribution. In many cases, Gaussian distribution can be approached even with
several segments if probability distribution function of each segment is
symmetric.

Once probability of end to end distance of a molecular chain is obtained,
entropy of this single chain is related to its micro-states by Boltzmann’s
entropy equation. If all molecular chains are taken to deform affinely with
the macroscopic deformation, probability distribution of a single molecular
chain can be extended to an entire polymer system. For an entire polymer
system, Boltzmann’s entropy equation can be written as

$S\>=\>k_{B}ln\Omega,$ (1.3)

where $S$ is entropy of an entire polymer system and $k_{B}=1.3807\times
10^{-23}JK^{-1}$ is Boltzmann’s constant. $\Omega$ is the number of micro-
states that this system can have. Originally developed for idea gas,
Boltzmann’s entropy equation was firstly introduced by Werner Kuhn in 1930s to
model rubber elasticity. This method was later adopted by Paul Flory and many
other researchers [14, 15, 16, 17]. This entropy concept still dominates
rubber elasticity up to nowadays. Boltzmann’s entropy equation can be derived
by using a statistical representation of temperature and the first law of
thermodynamics. If a system goes to equilibrium, it will select a macroscopic
configuration that maximizes the number of micro-states. If we assume that
there is only one single micro-state for each value of energy or no degeneracy
in the system, the number of micro-states $\Omega$ is proportional to
probability distribution of energy in the system. Therefore, the number of
micro-states $\Omega$ will depend on system energy which comes from each
micro-state of the system. Mathematically, $\Omega$ can be written as
$\Omega(E)$ where $E$ is system energy.

After entropy of the entire system is obtained, Helmholtz free-energy, $F$,
and entropy, $S$, are related by definition

$F=U-TS,$ (1.4)

where $U$ is internal energy and $T$ is temperature. Entropy elasticity
assumes that the change of Helmholtz free energy is mostly due to the change
of entropy. Therefore, internal energy can be approximately viewed as a
constant. Its contribution to force and stretch is nearly zero. By ignoring
the effect of $U$, Eq.(1.4) is changed to

$F=-TS.$ (1.5)

Combining Eq.(1.3) and Eq.(1.5), we obtained

$F=-k_{B}Tln\Omega.$ (1.6)

Helmholtz free energy is defined as free energy with constant temperature and
fixed volume. For most solids, these constraints are approximately true [18].
So strain energy density in solids defined as $W$ is considered to be the same
as Helmholtz free energy. With strain energy density given as Helmholtz free
energy, stress can be obtained by taking derivative of strain energy density
with respect to strain or stretch. If strain energy density is based on
Gaussian distribution of molecular chains, Neo-Hookean hyperleastic model can
be obtained. If strain energy density is based on Langevin chains, 3 chain
model, 4 chain model, and 8 chain model can be obtained. Gent model can be
shown to have a close relationship with 8 chain model [19]. 8 chain model or
Arruda-Boyce model [20] is widely used as a nonlinear spring for many large
deformation viscoelastic models [7, 21, 22]. Recently, many researchers work
on rubber elasticity by other non-Gaussian chain models [23, 24]. Their
mathematical structures are very complicate. But their ideas are still based
on entropy elasticity framework. To fulfill some applications or needs, many
other phenomenological hyperelastic models such as Ogden’s hyperelastic model
[25] or many $I_{1}$ based hyperelastic models [26] were also built by
assuming some form of strain energy density function to fit experimental data.
No statistical probability distributions of microscopic molecular chains are
considered in these models.

##  2 Deep Into Entropy Elasticity

Although entropy elasticity gains lots of success in explaining mechanical
behavior of rubbers, many concepts used in entropy elasticity need a second
thought. In entropy elasticity, Helmholtz free energy is related to the number
of micro-states by Eq.(1.6). As we know, the number of micro-states is very
hard to calculate. For example, given a molecular chain which has 100 links,
each link can elongate or contract which is assumed to have 2 micro-states.
Hence, there are $2^{100}=1.267E30$ micro-states. But this molecular chain has
only 101 macro-states. One marco-state can be elongation of all 100 links.
Another marco-state could be contraction of 1 link and elongation of the rest
99 links. Each macro-state is related to different number of micro-states. It
is almost impossible to find all micro-states. But it is possible to find the
probability of each macro-state. If we define the number of micro-states of a
macro-state $i$ to all number of micro-states as probability $P_{i}$, entropy
of the system can be given as Gibbs entropy equation

$S=-k_{B}\sum_{i}P_{i}lnP_{i}.$ (2.1)

By using Gibbs entropy equation, Helmholtz free energy is given as

$F=-k_{B}TlnZ,$ (2.2)

where $Z$ is partition function given as
$Z=\sum_{i}\mathbf{e}^{-E_{i}/{k_{B}T}}$. $E_{i}$ is the energy of each macro-
state. So we have two Helmholtz free energy equations, e.g., Eq. (1.6) and Eq.
(2.2). One is based on total micro-states and the other one is based on
partition function. In entropy elasticity, Gaussian probability function of
the end to end length vector of a molecular chain is found. Should it be
viewed as the probability distribution of each micro-state to total micro-
states? Or should it be viewed as the probability distribution of the number
of micro-states of a macro-state $i$ to all number of micro-states, e.g.,
$P_{i}$? This is not clear. Different consideration will lead to different
ways to calculation of Helmholtz free energy. Since Helmholtz free energy is
used as strain energy density for hyperelastic materials, different stress-
strain relationship will be got. When we solve conservation laws with
different constitutive models, it will lead to different results [27, 28, 29].

Fig. 1:  Schematic illustration of a molecular chain (orange color) in a relax
state (a) and in a stretched state (b).

The second concern about entropy elasticity is that its firstly dealing with
only one molecular chain by probability distribution. Then an affine or non-
affine [30] or phantom assumption [8] is made to extend to an entire molecular
chain system. In entropy elasticity, it assumes that a single molecular chain
can have more configurations in relax states than after stretched. Since it
assumes that these configurations are proportional to micro-states, micro-
states and entropy will be deduced after stretched. If a single molecular
chain is fully stretched to its contour length, it will have only one
configuration. Therefore, entropy is the smallest. This thought is mostly
based on freely jointed chain assumption. In freely jointed chains, each link
can freely rotate and is rigid. No internal energy change happens when a
freely jointed chain changes its configurations.

However, in real polymer materials, polymer chains are not freely jointed.
Rotation of each bond is associated with energy redistribution [31]. Bonds
between monomers are not rigid. Bond length is not constant. It can elongate
or contract under an external force. Bond length variation will be related to
energy change of the entire molecular chain.

Because bond number of a molecule chain is the same before stretched and after
stretched, micro-states of a molecular chain should remain the same even a
molecular chain has more configurations in a relaxed state. Since these micro-
states can be empty or occupied by electrons, the difference of a molecular
chain before stretched and after stretched is the probability redistribution
of electron occupancy in these micro-states. If more electrons occupy in
higher energy micro-states, entire energy of the system will be higher. That
is, after stretched by an external force, more high energy micro-states are
occupied by electrons in each bond of molecular chains. Therefore, more
configurations of a molecular chain do not necessarily mean more micro-states.
Entropy is not necessarily decreasing due to stretched chains.

In entropy elasticity, since entropy is related to micro-states by Boltzmann’s
entropy equation, internal energy is completely separated from entropy and
ignored. This is probably true for freely joint rigid chain. In fact, for real
chain, those electron filled micro-states will have internal energy. Those
empty micro-states will have no internal energy. Because entropy is related to
micro-states by Boltzmann entropy equation, entropy should be a function of
internal energy [32].

In entropy elasticity, stiffing effect in stress-strain curve is considered as
a molecular chain reaching its contour length or its full length. However,
even under large tensile force, a molecular chain can hardly reach its contour
length. This is because that there are many randomly distribution chains
instead of one molecular chain inside polymers. The stiffing effect in stress-
strain curve of rubber materials can be viewed as more entanglements formed
after being stretched, See Fig.(1). Before stretched, each molecular chain can
hardly feel the exist of other molecular chains. After putting tensile force
on it, molecular chains will move like snakes or reptations [2]. They will
start to have interactions with other molecular chains. Many entangled forces
will be built up after stretched. These gradually built-up entangled forces
are the cause of stiffing effect which is shown in rubber force-stretch curve.
It can be physically considered as more defects along a molecular chain. Or
some parts of a molecular chain get out of its reptation tube if a size of
reptation tube is assumed [33]. If force continues to increase, eventually the
molecular chain will break. Fracture will happen [34].

Another evidence that makes people to believe that entropy plays a role in
rubber elasticity is that rubbers become stiffer while metals become softer
after heating them up [35]. Flory called this force as the force of
retraction. When we observed an abnormal phenomena, there can be many
explanations. Entropy can be one of them. But we think it is due to
fluctuation of molecular chains instead of entropy change. Unlike crystalline
metals, molecular chains are very long with random distribution. With
temperature rising, each molecular chain can easily feel the other molecular
chains. More temporary bonds or dynamical bonds in transient networks [36, 24]
can be formed due to large amplitude vibrations of molecular chains in high
temperature. Therefore, bond density increases with temperature. This will
increase Young’s modulus of rubbers. But this change is small and temporary.
The overall change of Young’s modulus from glassy zone to rubbery zone is
decreasing [1].

Entropy elasticity is originally built by considering static loading with
constant temperature and fixed pressure. In fact, the contraints of Helmholtz
free energy are constant temperature and fixed volume. Hence, it’s hard for
existed entropy elasticity to extend to temperature and hydrostatic pressure
domain [37]. It is also not easy to extend to time domain. This is because
entropy elasticity is built on equilibrium thermodynamics [38]. First law of
thermodynamics tell us that energy will be conserved. but didn’t tell us how
fast and how slow the energy flows. To extend statistical elasticity to time,
temperature and hydrostatic pressure dependent area, we believe that it will
rely on how to describe bond deformations of molecular chains [39]. To
consider this, statistical description of covalent electrons should be used
since each bond is formed by covalent electrons. Borrowed from statistical
description of gas molecules, current entropy elasticity theory is not
appropriate to understand these bond movements. The big difference between gas
molecules and covalent electrons in solids is that the distance between two
gas molecules is very large comparing with the distance between two electrons.
Physically, the distance between two gas molecules is very larger than
”thermal” de Broglie wavelength. The moving path of one gas molecule can be
distinguished from that of another gas molecular. They are treated as
distinguishable particles. The distance between two covalent electrons is
comparable or smaller than ”thermal” de Broglie wavelength. They are fermions
and obey the Pauli exclusion principle. They are treated as indistinguishable
particles.

For distinguishable particles such as gas molecules, classical statistical
method such as Boltzmann’s entropy equation and Maxwell-Boltzmann distribution
are good candidates. For solids like rubbers, molecular chains are glued by
covalent electrons which are indistinguishable particles, quantum statistical
method such as Fermi-Dirac statistics need be used.

In the next section, we will show how we can go from Maxwell-Boltzmann
distribution to Fermi-Dirac distribution based on a molecular chain structure.

##  3 From Gaussian distribution to Maxwell-Boltzmann distribution to Fermi-
Dirac distribution

Firstly let us exam Eq.(1.1) in detail. One dimensional Gaussian distribution
for end to end distance of a molecular chain, e.g., Eq.(1.1) can be written as

$G(x,N)=\frac{\mathbf{e}^{-3x^{2}/2Nb^{2}}}{(\,\frac{3}{2\pi
Nb^{2}})\,^{1/2}}.$ (3.1)

or

$G(x,N)=\frac{\mathbf{e}^{-kx^{2}/2}}{L}$ (3.2)

if let $k=3/Nb^{2}$ and $L=(\,\frac{3}{2\pi Nb^{2}})\,^{1/2}$. Eq.(3.2) can be
further written as

$G(x,N)=P(\textrm{microstate in x
direction})=\frac{\mathbf{e}^{-E_{x}}}{\sum_{i}\mathbf{e}^{-E_{i}}}$ (3.3)

if let energy $E_{x}=kx^{2}/2$ and $L=\sum_{i}\mathbf{e}^{-E_{i}}$.

However, from classical thermodynamics, we know probability of micro-state $r$
can be written as

$P(\textrm{microstate
r})=\frac{\mathbf{e}^{-E_{r}/k_{B}T}}{\sum_{i}\mathbf{e}^{-E_{i}/k_{B}T}}$
(3.4)

where $\sum_{i}\mathbf{e}^{-E_{i}/k_{B}T}$ is partition function.

If comparing Eq.(3.3) with Eq.(3.4) , we know that $k_{B}T$ is missing in one
dimensional Gaussian distribution for end to end distance of a molecular
chain, e.g., Eq.(1.1). Should we add $k_{B}T$ into Gaussian distribution,
e.g., Eq.(1.1)? The answer is yes. In the following section, we will
illustrate the reason why $k_{B}T$ need be added.

Fig. 2:  Schematic illustration of vibration of monomers due to temperature
change. Vibration amplitude $A$ is proportional to $k_{B}T$.

If each bond segment length vector of a molecular chain is projected to
x-axis, segment length vector in x-direction, e.g., $X_{i}$, can be considered
as a random variable, the total chain length vector in $x$ direction can be
the summation of $X_{i}$, e.g., $S_{n}=X_{1}+...+X_{n}$. For each random
variable $X_{i}$, it measures the distance between two monomers in
x-direction, see Fig.(2). For all random variables $X_{i}$, let’s assume that
they are independent and have the same mean $\nu$ and the same standard
deviation $\sigma$. When temperature is increasing, monomer vibration
amplitude will increase. Assuming that all monomers vibrate in their
equilibrium positions, average distance between two monomers will remain the
same. But variation of the distance between two monomers will increase. In
other words, mean $\nu$ of $X_{i}$ remains the same while the standard
deviation $\sigma$ of all $X_{i}$ will increase with temperature. Even though
we don’t know exact probability distribution of $X_{i}$, central limit theorem
(A.2) tells us that total chain length vector will become Gaussian
distribution which can be written as

$Z_{n}=\frac{S_{n}-n\nu}{\sqrt{n}\sigma}.$ (3.5)

So the random variable $S_{n}$ which is used to measure the end-to-end length
vector of a molecular chain in x-direction is a normal distribution with mean
$n\nu$ and standard deviation $\sqrt{n}\sigma$. Since $\sigma$ will depend on
temperature $T$, we can select that $\sigma^{2}\varpropto k_{B}T$.

In real polymer materials, different molecular chains can have different bond
numbers. As long as unit number $n$ of a molecular chain is larger than a
critic number, all primary molecular chains will converge to normal
distribution. This critic number is usually very small if probability
distribution function of each bond length is symmetric. If primary molecular
chains are cross-linked, central limit theorem can also be used. As long as
the unit number of cross-linked molecular chain is large enough, summation of
random variables following each path of cross-linked molecular chain could be
viewed as normal distribution.

Let $\bar{\nu}$ be the average of all means of molecular chains. Let
$\bar{\sigma}$ be the average of all standard deviations of molecular chains
by defining $\bar{\sigma}^{2}=k_{B}T/k_{0}$. $k_{0}$ is a proportional
constant which is not necessarily equal to $k=3/Nb^{2}$. Then Gaussian
distribution function of $S_{n}$ can be written as

$G(x,N)=\frac{\sqrt{k_{0}}}{\sqrt{2\pi
k_{B}T}}\mathbf{e}^{-\frac{k_{0}}{2k_{B}T}\left(x-\bar{\nu}\right)^{2}}.$
(3.6)

By using equal partition theorem and let $\bar{\nu}\rightarrow 0$, Eq.(3.6)
becomes Maxwell-Boltzmann distribution in Cartesian coordinate system, e.g.,

$G(x,N)=\sqrt{\frac{m}{2\pi
k_{B}T}}\mathbf{e}^{-\frac{m{v_{x}}^{2}}{2k_{B}T}},$ (3.7)

where $m$ is mass of gas molecules and $v_{x}$ is velocity of gas molecules in
x-direction. Velocity distribution of gas molecules can be found by Maxwell-
Boltzmann distribution in polar coordinate system. Maxwell-Boltzmann
distribution can be viewed as temperature dependent Gaussian distribution.
Here, we use Maxwell-Boltzmann distribution to describe probability
distribution of the end-to-end length vector of a molecular chain with an
external force. When an external force is applied to the molecular chain,
electrons will move from low energy states to high energy states. Probability
distribution of the end-to-end length vector of a molecular chain can be
described by Maxwell-Boltzmann distribution. But energy storage inside the
deformed molecular chain is related to electron occupancy in each quantum
micro-states. The probability to find electron occupancy in each quantum
micro-state need be described by Fermi-Dirac function. Once we know the end-
to-end length vector of a molecular chain, energy storage due to electron
occupancy in each quantum micro-state can be found. In another word, we can
derive Fermi-Dirac function by using Maxwell-Boltzmann distribution, e.g.,
Eq.(3.6).

When an external force is applied to polymer materials, this force will do
work on polymer materials. Some of the work will become strain energy absorbed
by molecular chains. The configuration of molecular chains will get changed.
Because of the randomness of molecular chains, we don’t know how each bond
changes its rotation angle and bond length. We only know the probability
distribution of the end to end length vector of each polymer chain obeys
Maxwell-Boltzmann distribution.

Firstly let’s consider a signal transmission problem. If an unknown signal
(probable inputs are assumed to be 1/2 and -1/2) is sent to polymer materials,
this signal will go through these molecular chains. Measured output signal
will be contaminated by those Maxwell-Boltzmann distributed molecular chains,
which can be viewed as a random noise. Output signal is a combination of
random noise and input signal and becomes a continuous random signal. If we
can measure the output continuous random signal, probability to find which
input signal is sent with known measured output signal is given by mixed Bayes
rule.

Fig. 3:  Representation of a Gaussian molecular chain in two dimensions under
a random external force. $K$, $S_{n}$ and $Y$ are considered as random
variables.

Analogy to this signal problem, consider the length change of a molecular
chain under an input force. Let’s assume that an unknown force is applied at
the starting point of the chain, see Fig.(3). The displacement at the starting
point due to this unknown force is defined as a discrete random variable $K$.
Without losing generality, let probability to be 0.5 if this end moves to the
left by $-0.5$ and probability to be 0.5 if it moves to the right by $0.5$.
That is, the discrete random variable $K$ is defined as

$P_{K}(k)=\begin{cases}0.5,\text{if $k=0.5$}\\\ 0.5,\text{if
$k=-0.5$}\end{cases}$ (3.8)

Recall that $S_{n}$ is a continuous random variable of the end to end length
vector of a molecular chain with an external force, see Fig.(3). After
imposing a force on the starting point of a molecular chain, we want to locate
the ending point of the molecular chain. Let’s define a continuous random
variable $Y$ which measures the total length from the ending point of this
deformed molecular chain to the starting point of this initial undeformed
chain, See Fig.(3). So mathematically $Y=K+S_{n}$. That is, total length is
the combination of input displacement and end to end length of a molecular
chain.

Firstly, let’s consider two extreme cases when molecular chains are extremely
stiff or extremely soft. If a molecular chain is super stiff, variation of
each bond length will be close to $0$. Temperature dependent Gaussian
distribution (Maxwell-Boltzmann distribution) of a molecular chain $S_{n}$
will collapse to delta function with constant mean and variance is nearly $0$.
For example, assuming an input displacement $K$ is $1/2$ and the length of a
molecular chain $S_{n}$ is $10$, output length $Y$ will be $10.5$. In other
words, the probability of finding input displacement is $1/2$ given measured
output length $10.5$ is $1$. It also says, the probability that available
micro-states are occupied by electrons in this super stiff molecular chain is
$1$.

If a molecular chain is super soft, variance of temperature dependent Gaussian
distribution will approach infinity. Probability distribution of the end to
end distance vector of a molecular chain can be viewed as uniform distribution
from negative infinity to positive infinity. Hence, it is hard to determine
mean value of $S_{n}$. Assuming input displacement is $1/2$ and the length of
a molecular chain is 10, output length can not be determined at all because
the variance of this super soft chain is infinity. In other words, the
probability of finding input displacement is $1/2$ given any measured output
length is $0$. This gives us an idea that available micro-states are all empty
in this super soft molecular chain. Or the probability that available micro-
states are occupied by electrons is $0$.

Between those two extreme cases when a molecular chain obeys temperature
dependent Gaussian distribution, probability of available micro-states
occupied by electrons is between $0$ and $1$.

Therefore, probability to find input displacement, e.g., $K$, imposed on a
molecular chain with given measured total length of a deformed molecular chain
$Y$ will be the probability that available micro-states are occupied by
electrons.

Even though we consider only one molecular chain configuration here, Eq. (3.6)
is applied to all molecular chains since average mean and average variance are
used.

By using mixed Bayes rule, the probability of finding input displacement $k$
given any measured output length $y$ is given as posterior probability
distribution, e.g., $P_{K|Y}(k|y)$. It also tells us the probability that
available micro-states are occupied by electrons. To obtain the posterior
probability distribution, e.g., $P_{K|Y}(k|y)$, prior probability distribution
$f_{Y|K}(y|k)$ and $f_{Y}(y)$ need be calculated firstly. Since the random
variable of measured output length $Y=K+S_{n}$, prior probability distribution
$f_{Y|K}(y|k)$ is a conditional Maxwell-Boltzmann distribution. That is, if we
know the exact displacement at the starting point of a molecular chain, the
probability to find the other end of a molecular chain is given by conditional
Maxwell-Boltzmann distribution $f_{Y|K}(y|k)$. $f_{Y}(y)$ is the probability
to find the other end of a molecular chain regardless of input displacement.

A brief review of mixed Bayes rule is given in the appendix. Bayesian
inference has also been used to evaluate parameters of constitutive models and
polymer properties [40, 41].

To proceed, the conditional Maxwell-Boltzmann distribution $f_{Y|K}(y|k)$ can
be written as

$f_{Y|K}(y|k)\>=\>\sqrt{\frac{k_{0}}{2\pi
k_{B}T}}e^{-\frac{k_{0}}{2k_{B}T}(y-\bar{\nu}-k)^{2}}.$ (3.9)

In order to use mixed Bayes rule, $f_{Y}(y)$ is calculated by

$f_{Y}(y)\>=\>\frac{1}{2}\sqrt{\frac{k_{0}}{2\pi
k_{B}T}}e^{-\frac{k_{0}}{2k_{B}T}(y-\bar{\nu}-\frac{1}{2})^{2}}+\frac{1}{2}\sqrt{\frac{k_{0}}{2\pi
k_{B}T}}e^{-\frac{k_{0}}{2k_{B}T}(y-\bar{\nu}+\frac{1}{2})^{2}}.$ (3.10)

Substituting Eq.(3.8), Eq.(3.9) and Eq. (3.10) into Eq.(A.7), we can get the
probability of input distance is $-1/2$ if given measured total distance $y$
as

$P_{K|Y}(-\frac{1}{2}|y)\>=\>\frac{1}{e^{\frac{k_{0}(y-\bar{\nu})}{k_{B}T}}+1}.$
(3.11)

By letting $\varepsilon=k_{0}y$ and $\mu=k_{0}\bar{\nu}$, Eq.(3.11) is recast
as

$P_{K|Y}(-\frac{1}{2}|y)\>=\>\frac{1}{e^{\frac{\varepsilon-\mu}{k_{B}T}}+1},$
(3.12)

which is Fermi-Dirac distribution function. Fermi Dirac distribution is the
probability that an available energy micro-state at energy level $\varepsilon$
is occupied by an electron. $\mu$ is called chemical potential or Fermi
energy. Fermi-Dirac distribution can be used to calculate energy storage in
each bond of a molecular chain if we know the density of micro-states.
Similarly, the probability of input distance is $1/2$ if given measured total
distance $y$ is given as

$P_{K|Y}(\frac{1}{2}|y)\>=\>\frac{1}{e^{-\frac{\varepsilon-\mu}{k_{B}T}}+1},$
(3.13)

which is the probability of finding a hole at an available energy micro-state.
The probability of finding an electron in an available micro-state is one
minus the probability of finding a hole in an available micro-state, e.g.,

$\frac{1}{e^{\frac{\varepsilon-\mu}{k_{B}T}}+1}\>=\>1-\frac{1}{e^{-\frac{\varepsilon-\mu}{k_{B}T}}+1}.$
(3.14)

In polymer materials, cross-links and entanglements are co-existed. As we
discussed, cross-links are bonds formed by covalent electrons, which can be
modeled by Fermi-Dirac distribution. Entanglements need be considered
separately. They are weak forces compared to forces caused by covalent bonds.
Entanglements can be viewed as a transient network. Density and velocity of
entanglements can be easily changed by an external force. A possible treatment
of entanglements is considering them as defects along molecular chains [2]. In
another word, defect density and defect velocity will change under an external
force. Idea of this treatment is consistent with tube and reptation models
[33]. It will be treated as a defect if any part of a molecular chain gets out
of the assumed tube under an external force. The mathematical structure of
Fermi-Dirac distribution is also much easier and cleaner than the slip-link
model introduced by Edwards and Vilgis [42, 43].

Mathematically, Fermi-Dirac distribution function is logistic function.
Logistic function can be generalized to include time, temperature, hydrostatic
pressure effects. By putting generalized Fermi-Dirac distribution into our
stress-strain equation, many mechanical behavior problems of polymers can be
easily understood. For example, shear band and necking formation is localized
plastic deformation if defect effect is larger than bonding effect [3].
Ultrasonic vibration is considered to be an inverse process of shear band
generation [4]. Knee injury can be prevented by erasing physical aging which
can be understood by time dependent version of generalized Fermi-Dirac
distribution [44]. Generalized logistic function has been used to model linear
viscoelasticity [1]. Time, temperature, and hydrostatic pressure dependent
viscosity of polymers can be understood by generalized Fermi-Dirac
distribution. Aided by defect concepts in molecular chains, modeling nonlinear
viscoplasiticty of polymers is straightforward [2]. Model prediction is
validated by many experiments shown in our previous papers[2, 1].

##  4 Conclusion

In this article, we showed how to derive from temperature independent Gaussian
distribution to temperature dependent Gaussian distribution or Maxwell-
Boltzmann distribution. Then Fermi-Dirac distribution is derived from Maxwell-
Boltzmann distribution based on a deformed molecular chain under an external
force. Since the distance between gas molecules is very larger than ”thermal”
de Broglie wavelength, Maxwell-Boltzmann distribution should be used for
describing the energy distribution for gas molecules. On the other hand,
Fermi-Dirac distribution is needed to capture energy distribution for covalent
bonds in polymer molecular chains. Coupled with defects or entanglements along
molecular chains, generalized Fermi-Dirac distribution can be used to explain
most if not all polymer mechanical problems. Since human tissue is made of
molecular chains of different bond stiffness, this method will be extended to
explain intra-cranial brain tissue dynamics in the next paper.

## Data Availability

Data sharing not applicable - no new data generated.

## Reference

## References

  * [1] L. Yang, L. Yang, R. L. Lowe, A viscoelasticity model for polymers: Time, temperature, and hydrostatic pressure dependent Young’s modulus and Poisson’s ratio across transition temperatures and pressures, Mechanics of Materials 157 (2021) 103839.
  * [2] L. Yang, A Mathematical Model for Amorphous Polymers Based on Concepts of Reptation Theory, Polymer Engineering & Science 59 (11) (2019) 2335–2346.
  * [3] L. Yang, L. Yang, Revisit initiation of localized plastic deformation: Shear band & necking, Extreme Mechanics Letters 40 (2020) 100914.
  * [4] L. Yang, L. Yang, Digest Ultrasonic Welding I: Localized Heating and Fuse Bonding, arXiv preprint arXiv:2301.05810 (2023) 1–13.
  * [5] L. Zhan, S. Wang, S. Qu, P. Steinmann, R. Xiao, A new micro–macro transition for hyperelastic materials, Journal of the Mechanics and Physics of Solids 171 (2023) 105156.
  * [6] C. Li, Z. Wang, Y. Wang, Q. He, R. Long, S. Cai, Effects of network structures on the fracture of hydrogel, Extreme Mechanics Letters 49 (2021) 101495.
  * [7] Q. Wang, G. R. Gossweiler, S. L. Craig, X. Zhao, Mechanics of mechanochemically responsive elastomers, Journal of the Mechanics and Physics of Solids 82 (2015) 320–344.
  * [8] Y. Akagi, T. Katashima, Y. Katsumoto, K. Fujii, T. Matsunaga, U.-i. Chung, M. Shibayama, T. Sakai, Examination of the Theories of Rubber Elasticity Using an Ideal Polymer Network, Macromolecules 44 (14) (2011) 5817–5821.
  * [9] M. R. Buche, M. N. Silberstein, Chain breaking in the statistical mechanical constitutive theory of polymer networks, Journal of the Mechanics and Physics of Solids 156 (2021) 104593.
  * [10] F. Zhang, S. Cui, A two-component statistical model for natural rubber, Polymer 242 (2022) 124462.
  * [11] F. Zhang, Z. Gong, W. Cai, H.-j. Qian, Z.-y. Lu, S. Cui, Single-chain mechanics of cis-1,4-polyisoprene and polysulfide, Polymer 240 (2022) 124473.
  * [12] P. J. Flory, Principles of Polymer Chemistry, Cornell University Press, 1953\.
  * [13] E. Darabi, M. Hillgärtner, M. Itskov, An amended approximation of the non-Gaussian probability distribution function, Mathematics and Mechanics of Solids 28 (2) (2023) 521–532.
  * [14] F. Tanaka, S. F. Edwards, Viscoelastic properties of physically crosslinked networks. 1. Transient network theory, Macromolecules 25 (5) (1992) 1516–1523.
  * [15] F. J. Vernerey, R. Brighenti, R. Long, T. Shen, Statistical damage mechanics of polymer networks, Macromolecules 51 (17) (2018) 6609–6622.
  * [16] R. Mahnken, J. Mirzapour, A statistically based strain energy function for polymer chains in rubber elasticity, Archive of Applied Mechanics 92 (11) (2022) 3295–3323.
  * [17] Y. Jiang, L. Li, Y. Hu, Strain gradient elasticity theory of polymer networks, Acta Mechanica 233 (8) (2022) 3213–3231.
  * [18] X. Zhao, A theory for large deformation and damage of interpenetrating polymer networks, Journal of the Mechanics and Physics of Solids 60 (2) (2012) 319–332.
  * [19] L. Yang, L. Yang, Note on Gent’s hyperelastic model, Rubber Chemistry and Technology 91 (1) (2018) 296–301.
  * [20] E. M. Arruda, M. C. Boyce, A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials, Journal of the Mechanics and Physics of Solids 41 (2) (1993) 389–412.
  * [21] Y. Jiang, L. Li, Y. Hu, A physically-based nonlocal strain gradient theory for crosslinked polymers, International Journal of Mechanical Sciences 245 (2023) 108094\.
  * [22] M. Uchida, K. Kamimura, T. Yoshida, Y. Kaneko, Viscoelastic-viscoplastic modeling of epoxy based on transient network theory, International Journal of Plasticity 153 (2022) 103262.
  * [23] S. A. Mezzasalma, M. Abrami, G. Grassi, M. Grassi, Rubber elasticity of polymer networks in explicitly non-Gaussian states. Statistical mechanics and LF-NMR inquiry in hydrogel systems, International Journal of Engineering Science 176 (2022) 103676.
  * [24] F. J. Vernerey, R. Long, R. Brighenti, A statistically-based continuum theory for polymers with transient networks, Journal of the Mechanics and Physics of Solids 107 (2017) 1–20.
  * [25] G. A. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science, Meccanica 37 (4) (2002) 489–490.
  * [26] O. Lopez-Pamies, A new I1-based hyperelastic model for rubber elastic materials, Comptes Rendus Mécanique 338 (1) (2010) 3–11.
  * [27] S.-T. J. Yu, L. Yang, R. L. Lowe, S. E. Bechtel, Numerical simulation of linear and nonlinear waves in hypoelastic solids by the CESE method, Wave Motion 47 (3) (2010) 168–182.
  * [28] Y.-Y. Chen, L. Yang, S.-T. J. Yu, Simulations of waves in elastic solids of cubic symmetry by the conservation element and solution element method, Wave Motion 48 (1) (2011) 39–61.
  * [29] L. Yang, Y.-Y. Chen, S.-T. J. Yu, Viscoelasticity determined by measured wave absorption coefficient for modeling waves in soft tissues, Wave Motion 50 (2) (2013) 334–346.
  * [30] C. Miehe, S. Göktepe, F. Lulei, A micro-macro approach to rubber-like materials—Part I: the non-affine micro-sphere model of rubber elasticity, Journal of the Mechanics and Physics of Solids 52 (11) (2004) 2617–2660.
  * [31] J. Mulderrig, B. Talamini, N. Bouklas, A statistical mechanics framework for polymer chain scission, based on the concepts of distorted bond potential and asymptotic matching (Aug. 2022).
  * [32] Y. Mao, B. Talamini, L. Anand, Rupture of polymers by chain scission, Extreme Mechanics Letters 13 (2017) 17–24.
  * [33] Y. Li, S. Tang, B. C. Abberton, M. Kröger, C. Burkhart, B. Jiang, G. J. Papakonstantopoulos, M. Poldneff, W. K. Liu, A predictive multiscale computational framework for viscoelastic properties of linear polymers, Polymer 53 (25) (2012) 5935–5952.
  * [34] J. Kim, G. Zhang, M. Shi, Z. Suo, Fracture, fatigue, and friction of polymers in which entanglements greatly outnumber cross-links, Science 374 (6564) (2021) 212–216.
  * [35] Y. Yoshikawa, N. Sakumichi, U.-i. Chung, T. Sakai, Negative energy elasticity in a rubberlike gel, Physical Review X 11 (1) (2021) 011045.
  * [36] T. Katashima, R. Kudo, M. Naito, S. Nagatoishi, K. Miyata, U.-i. Chung, K. Tsumoto, T. Sakai, Experimental Comparison of Bond Lifetime and Viscoelastic Relaxation in Transient Networks with Well-Controlled Structures, ACS Macro Letters 11 (6) (2022) 753–759.
  * [37] T. Katashima, R. Kobayashi, S. Ishikawa, M. Naito, K. Miyata, U.-i. Chung, T. Sakai, Decoupling between Translational Diffusion and Viscoelasticity in Transient Networks with Controlled Network Connectivity, Gels 8 (12) (2022) 830.
  * [38] H. Zhang, Y. Hu, A statistical-chain-based theory for dynamic living polymeric gels with concurrent diffusion, chain remodeling reactions and deformation, Journal of the Mechanics and Physics of Solids 172 (2023) 105155.
  * [39] M. R. Buche, M. N. Silberstein, S. J. Grutzik, Freely jointed chain models with extensible links, Physical Review E 106 (2) (2022) 024502.
  * [40] A. J. Thomas, E. Barocio, I. Bilionis, R. B. Pipes, Bayesian inference of fiber orientation and polymer properties in short fiber-reinforced polymer composites, Composites Science and Technology 228 (2022) 109630.
  * [41] T. Long, S. Shende, C.-Y. Lin, K. Vemaganti, Experiments and hyperelastic modeling of porcine meniscus show heterogeneity at high strains, Biomechanics and Modeling in Mechanobiology 21 (6) (2022) 1641–1658.
  * [42] J. D. Davidson, N. C. Goulbourne, A nonaffine network model for elastomers undergoing finite deformations, Journal of the Mechanics and Physics of Solids 61 (8) (2013) 1784–1797.
  * [43] S. F. Edwards, T. Vilgis, The effect of entanglements in rubber elasticity, Polymer 27 (4) (1986) 483–492.
  * [44] L. Yang, Theoretical and numerical analysis of anterior cruciate ligament injury and its prevention, Global Journals of Research in Engineering 20 (J1) (2020) 43–54.

##  Appendix A Brief review of central limit theorem and mixed Bayes rule

The basic idea of central limit theorem is that the summation of many
individual independent random variables will converge to the same Gaussian
distribution. Let $X_{1},...,X_{n}$ are $n$ identical independent distributed
random variables with finite mean $\nu$ and finite variance $\sigma^{2}$.
These $n$ random variables need not be normal distribution. $S_{n}$ is defined
as the sum of these $n$ identical independent distributed random variables,
e.g., $S_{n}=X_{1}+...+X_{n}$. A new random variable can be defined as

$Z_{n}=\frac{S_{n}-n\nu}{\sqrt{n}\sigma}.$ (A.1)

If $Z$ is a standard normal random variable with zero mean and unit variance,
central limit theorem tell us that $Z_{n}$ will be approach to $Z$ if $n$ goes
to infinity. Mathematically central limit theorem is expressed as

$Z_{n}=\lim_{n\to\infty}P(Z_{n}\leq z)=P(Z\leq z).$ (A.2)

Next, we will review mixed Bayes rule which mixes a discrete random variable
and a continuous random variable. Assume that we have a discrete random
variable $K$ and a continuous random variable $Y$. For the discrete random
variable $K$, we can define probability mass function as

$P_{K}(k)\>=\>P(K=k).$ (A.3)

Conditional probability mass function for discrete random variable $K$ given
continuous random variable $Y$ taking a value $y$ is defined as

$P_{K|Y}(k|y)\>=\>P(K=k|y\leq Y\leq y+\delta),$ (A.4)

where $\delta$ is a small increment. For the continuous random variable $Y$,
probability density function is defined as

$f_{Y}(y)\delta\>=\>P(y\leq Y\leq y+\delta).$ (A.5)

Conditional probability density function for continuous random variable $Y$
given discrete random variable $K$ taking a value $k$ is defined as

$f_{Y|K}(y|k)\delta\>=\>P(y\leq Y\leq y+\delta|K=k).$ (A.6)

So mixed Bayes rule can be written as

$P_{K|Y}(k|y)\>=\>\frac{P_{K}(k)f_{Y|K}(y|k)}{f_{Y}(y)},$ (A.7)

where

$f_{Y}(y)\>=\>\sum_{k^{\prime}}p_{K}(k^{\prime})f_{Y|K}(y|k^{\prime}).$