∎

11institutetext:  Eindhoven University of Technology, Department of Mechanical
Engineering, Polymer Technology, PO Box 513, NL–5600 MB Eindhoven, The
Netherlands. 11email<EMAIL_ADDRESS>22institutetext: Georgetown University,
Department of Physics, Institute for Soft Matter Synthesis and Metrology, 37th
and O Streets, N.W., Washington, D.C. 20057, USA. 22email:
<EMAIL_ADDRESS>33institutetext: School of Mathematics,
University of Leeds, Leeds LS2 9JT, U.K. 33email<EMAIL_ADDRESS>
# Fluctuating viscoelasticity based on a finite number of dumbbells

Markus Hütter  Peter D. Olmsted  Daniel J. Read

(To be published in European Physical Journal E, Accepted 2 November 2020)

###### Abstract

Two alternative routes are taken to derive, on the basis of the dynamics of a
finite number of dumbbells, viscoelasticity in terms of a conformation tensor
with fluctuations. The first route is a direct approach using stochastic
calculus only, and it serves as a benchmark for the second route, which is
guided by thermodynamic principles. In the latter, the Helmholtz free energy
and a generalized relaxation tensor play a key role. It is shown that the
results of the two routes agree only if a finite-size contribution to the
Helmholtz free energy of the conformation tensor is taken into account. Using
statistical mechanics, this finite-size contribution is derived explicitly in
this paper for a large class of models; this contribution is non-zero whenever
the number of dumbbells in the volume of observation is finite. It is noted
that the generalized relaxation tensor for the conformation tensor does not
need any finite-size correction.

###### Keywords:

Thermal fluctuations – Viscoelastic fluids – Dumbbell model – Conformation
tensor – Finite-size effects

##  1 Introduction

Fluctuations are particularly important when studying small systems. This also
holds for fluids, including complex fluids, e.g., macromolecular and polymeric
liquids. Small scales are involved, e.g., in microrheology squiresmason2010
and micro- and nanofluidic devices micro1 ; micro2 . For Newtonian fluids,
i.e., fluids with a deformation-independent viscosity and a lack of memory,
the dynamics on small scales could be described in terms of the fluctuating
Newtonian fluid dynamics developed by Landau and Lifshitz LL1959 . However,
this is not sufficient for complex fluids, and thus extensions are needed. For
example, the stress tensor has been related to the rate-of-strain tensor by a
memory kernel, and correspondingly colored noise has been introduced on the
stress tensor hohenegger2017 ; hohenegger2017addtl . Another approach towards
modeling fluctuating effects in complex fluids has been taken by Vázquez-
Quesada, Ellero, and Español vazquezquesadaetal2009 and applied to
microrheology vazquezquesadaetal2012 , in which smoothed-particle
hydrodynamics is extended by a conformation tensor that describes the
conformation of the small number of polymer chains per volume element. The
concept of fluctuating dynamics for the conformation tensor has been extended
recently HHA2018 , to make it applicable not only to the Maxwell model
berisedwards1990b ; wapperomhulsen1998 , as in vazquezquesadaetal2009 ;
vazquezquesadaetal2012 , but to a wider class of models, e.g. the FENE-P model
wapperomhulsen1998 ; wedgewoodbird1988 and the Giesekus model
wapperomhulsen1998 ; Giesekus1982RA ; Giesekus1982JNNFM ; birdwiest1985 . In
the approach taken in HHA2018 , the Helmholtz free energy in terms of the
conformation tensor plays an essential role.

The dynamics of the conformation tensor roots in a finer description, in
particular, it can be related to the kinetic theory of dumbbells (e.g., see
chapter 13 in birdbook ). The question addressed in this paper is what lessons
can be learned from deriving the dynamics for the conformation tensor with
fluctuations from an underlying kinetic description for a finite number of
dumbbells. It is pointed out that the dumbbell description already contains
the relaxation and fluctuation effects that are relevant also on the
conformation-tensor level. This is in contrast to coarse graining from an
atomistic description to bead-spring chains or directly to the conformation
tensor, e.g., see the work of Underhill and Doyle underhilldoyle2004 and of
Ilg et al. ilg2008 ; ilg2009 , respectively, without fluctuations on the
conformation-tensor level.

The paper is organized as follows. In sect. 2, a certain class of kinetic
dumbbell models is introduced, based on which a description of fluctuating
viscoelasticity in terms of the conformation tensor is derived via a direct
route, for a finite number of dumbbells. This route is paralleled in sect. 3,
where a thermodynamic approach is taken to arrive at fluctuating
viscoelasticity. As part of that, the finite-size correction to the Helmholtz
free energy is calculated, and this is found to be essential for finding
agreement between the two approaches. In the appendix, three different
calculation methods for this free energy are detailed, each of which arrives
at the same result. In sect. 4, the relation between the dumbbell models and
the multiplicative decomposition of the conformation tensor, which has been
discussed recently in the literature HHA2018 ; publ72 ; HCHA2020 , is
examined. The paper ends with conclusions and a discussion in sect. 5.

Throughout this paper, the following notation will be used. All summations are
spelled out, i.e., no Einstein summation-convention is used for repeated
indices. While the symbol $\cdot$ denotes a contraction of one pair of
indices, we use $\odot$ for a double contraction: For an order-four tensor
$\bm{A}^{(4)}$ and an order-two tensor $\bm{B}$,
$\left[\bm{A}^{(4)}\odot\bm{B}\right]_{ij}=\sum_{k,l}A^{(4)}_{ijkl}{B}_{kl}$.
The Kronecker delta is given as $\delta_{ij}$, and the Dirac delta-function as
$\delta(x-y)$. The dyadic product of two vectors $\bm{v}_{1}$ and $\bm{v}_{2}$
is written as $\bm{v}_{1}\bm{v}_{2}$.

##  2 Fluctuating viscoelasticity derived from dumbbell models

###  2.1 Dumbbell models

Let us consider $N$ dumbbells described by connector vectors $\bm{Q}_{\mu}$,
$\mu=1,\ldots,N$, in $D$-dimensional space. For the purpose of this study, the
following class of models is examined (see also birdbook ; hcostochbook ;
generic2 ; hcobook ),

$\displaystyle{\rm{d}}\bm{Q}_{\mu}$ $\displaystyle=$
$\displaystyle\left[\bm{\kappa}\cdot\bm{Q}_{\mu}-\bm{M}\cdot\frac{\partial\Phi}{\partial\bm{Q}_{\mu}}+k_{\rm{B}}T\left(\frac{\partial}{\partial\bm{Q}_{\mu}}\cdot\bm{M}\right)\right]{\rm{d}}t$
(1) $\displaystyle+\sqrt{2k_{\rm{B}}T}\bm{B}\cdot{\rm{d}}\bm{W}_{\mu}\,,$

which is a set of coupled stochastic differential equations (SDE) gardinerbook
; hcostochbook ; the Itô interpretation of stochastic calculus will be used
throughout this paper gardinerbook ; hcostochbook . In eq. (1),
$\bm{\kappa}=\partial\bm{v}/\partial\bm{r}$ is the gradient of the imposed
velocity field, $\Phi$ is the potential energy of all dumbbells, $\bm{M}$ is
the mobility tensor, with $\bm{M}^{\rm{T}}=\bm{M}$, $T$ stands for the
absolute temperature, $\bm{B}$ must satisfy the fluctuation-dissipation
theorem Groot ; Kubo ; Evans ; hcobook ,

$\displaystyle\bm{B}\cdot\bm{B}^{\rm{T}}=\bm{M}\,,$ (2)

and the vectors ${\rm{d}}\bm{W}_{\mu}$ are increments of independent Wiener
processes gardinerbook ; hcostochbook , satisfying

$\displaystyle\langle{\rm{d}}\bm{W}_{\mu}(t)\rangle$ $\displaystyle=$
$\displaystyle\bm{0}\,,\quad\forall\mu\,,$ (3)
$\displaystyle\langle{\rm{d}}\bm{W}_{\mu}(t){\rm{d}}\bm{W}_{\nu}(t^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\delta_{\mu\nu}\,\delta(t-t^{\prime})\,dt\,dt^{\prime}\bm{1}\,,\quad\forall\mu,\nu\,.$
(4)

It is noted that, in general, $\bm{B}$ has dimensions $D\times P$ with $P\geq
D$, and therefore ${\rm{d}}\bm{W}_{\mu}$, for any dumbbell $\mu$, has
dimensions $P\times 1$, i.e., is a $P$-dimensional vector for every particle
$\mu$. For practical purposes, one may choose $P=D$, although other choices
may also be convenient HHA2018 .

It can be shown that the probability distribution for the system (1) at
equilibrium is given by the Boltzmann distribution with energy $\Phi$; to
ensure that, the third term in the square brackets in eq. (1) is essential –
if this term was absent, the equilibrium distribution would depend on the
mobility tensor, which is unphysical. Note that the mobility $\bm{M}$ may
depend on $\\{\bm{Q}_{\mu}\\}_{\mu=1,\ldots,N}$, however, it is assumed in
this study that it is the same mobility tensor for all dumbbells $\mu$, and
thus it has no subscript $\mu$.

For completeness, it is mentioned that all position vectors are dimensionless,
i.e., they are scaled with respect to (w.r.t.) a characteristic constant
length-scale, that is omitted throughout this paper for convenience.

In the following, expressions for $\Phi$ and $\bm{M}$ are considered that will
allow us to eventually derive a closed evolution equation for the
instantaneous conformation tensor

$\displaystyle\hat{\bm{c}}=\frac{1}{N}\sum^{N}_{\mu=1}\bm{Q}_{\mu}\bm{Q}_{\mu}\equiv\overline{\bm{Q}\bm{Q}}\,.$
(5)

Symmetry requires that the potential energy $\Phi_{\rm d}$ of a single
dumbbell depends only on the (squared) length of the dumbbell vector. If the
dumbbells are not interacting, the total potential energy of all dumbbells is
given by the sum of the individual dumbbell contributions, i.e., $\Phi_{\rm
exact}=N\overline{\Phi_{\rm d}({\rm tr}\left(\bm{Q}\bm{Q}\right))}$. However,
in this paper, we consider models for which the total potential energy is
obtained by interchanging the operations $\overline{(\ldots)}$ and $\Phi_{\rm
d}({\rm tr}\left(\ldots\right))$ in $\Phi_{\rm exact}$, i.e., we use the
relation

$\displaystyle\Phi=N\Phi_{\rm d}({\rm{tr}\,}{\hat{\bm{c}}})\,,$ (6)

where eq. (5) was employed. This mean-field ansatz will be beneficial for
deriving closed dynamics for $\hat{\bm{c}}$. Obviously, eq. (6) is — for all
but one case (see below) — only an approximation to the exact potential for
all dumbbells. However, for the purpose of this paper (which is to examine the
effect of finite $N$ on the counting of states, and ramifications thereof for
the dynamics of $\hat{\bm{c}}$), we employ eq. (6) for defining the class of
models we examine. Therefore, we will not go into details about how accurately
eq. (6) approximates the exact potential.

The energy $\Phi$ given by eq. (6) equals the exact energy $\Phi_{\rm exact}$
only if the function $\Phi_{\rm d}({\rm tr}\left(\ldots\right))$ is linear, in
particular for linear-elastic Hookean springs, $\Phi_{\rm d}({\rm
tr}\left(\bm{Q}\bm{Q}\right))=(H/2){\rm tr}\left(\bm{Q}\bm{Q}\right)$, with
$H$ the spring constant of the dumbbell. However, other cases are of interest
as well. The dumbbell force generally has the form $-Hf\bm{Q}_{\mu}$. For
example, for a linear-elastic Hookean spring, $f=1$, while for a finitely
extensible nonlinear elastic spring one can use 111Sometimes a different
parametrization is used, e.g. $\beta_{1}=(b+3)/b$ and
$\beta_{2}=bH/k_{\rm{B}}T$, with dimensionless parameter $b$ HHA2018 . (see
also wedgewoodbird1988 ),

$\displaystyle
f=\left(\beta_{1}-\frac{H}{\beta_{2}k_{\rm{B}}T}{\rm{tr}\,}{\hat{\bm{c}}}\right)^{-1}\,,$
(7)

with two constants $\beta_{1}$ and $\beta_{2}$, and which accounts for the
finite extensibility in a mean-field sense. Typically, in the FENE-P
approximation wedgewoodbird1988 (see also peterlin1966 ), $\hat{\bm{c}}$ in
eq. (7) is used for infinitely many dumbbells. However, since in this paper
the focus is on studying systems with a finite number $N$ of dumbbells, we
generalize this by using eq. (7) for finite $N$. The potential that
corresponds to the desired expression for the force is given by

$\displaystyle\Phi=\frac{N\beta_{2}k_{\rm{B}}T}{2}\ln f\,,$ (8)

of which the limit $\beta_{1}=1$ and $\beta_{2}\to\infty$ results in the
potential for the linear-elastic Hookean spring. Therefore, using the
approximation $\Phi$, given by eq. (8) with eq. (7), instead of the exact
energy $\Phi_{\rm exact}$ is safe for small deformations. For other
deformations, it is noted that the potential $\Phi$ does include finite
extensibility, albeit in a different way than if applied to each dumbbell
individually. It is a topic of future research to examine the foundations of
the mean-field approximation $\Phi$ for finite $N$ thoroughly. In this paper,
this approximate expression for the energy forms part of defining the kinetic
models that are examined in the following, and it is suitable for deriving
closed dynamics for $\bm{c}$, in the same spirit as the FENE-P approximation
has been introduced earlier for infinite $N$ wedgewoodbird1988 .

As far as the mobility tensor $\bm{M}$ is concerned, its dependence on the
dumbbell vectors $\\{\bm{Q}_{\mu}\\}_{\mu=1,\ldots,N}$ is restricted to a
dependence on the instantaneous conformation tensor,

$\displaystyle\bm{M}=\bm{M}(\hat{\bm{c}})\,.$ (9)

A particular realization of that is

$\displaystyle\bm{M}$ $\displaystyle=$
$\displaystyle\frac{2}{\zeta}\left((1-\alpha)\bm{1}+\alpha\hat{\bm{c}}\right)\,,$
(10)

with friction coefficient $\zeta$, and the parameter $\alpha$ is used to
adjust the amount of $\hat{\bm{c}}$-dependence. In case of imposed
deformation, a non-zero value for $\alpha$ thus results in anisotropy of the
friction tensor, where the anisotropy is introduced in a mean-field sense. The
form (10) corresponds to the widely used Giesekus model for anisotropic drag
Giesekus1982RA ; Giesekus1982JNNFM ; birdwiest1985 . For the Giesekus model,
one typically uses eq. (10) for infinitely many chains, i.e., $N\to\infty$, to
render the model solvable. However, the mobility of a dumbbell is affected
primarily by the other dumbbells in its vicinity, and therefore the finite-$N$
generalization (10) is reasonable.

Allowing for the dumbbell potential energy $\Phi_{\rm d}({\rm
tr}\left(\ldots\right))$ in eq. (6) to be nonlinear and/or the dumbbell
mobility tensor $\bm{M}$ to depend on the conformation tensor effectively
introduces mean-field type couplings of the individual dumbbells. In practice
this implies that either (a) the $N$ dumbbells must be in the vicinity of each
other so they can interact, or (b) they diffuse rapidly enough in space to
effect such interactions. In the case of the mobility tensor, the implied
physics is reasonably clear: the assumption is that the average orientation of
surrounding dumbbells affects the mobility of any given test dumbbell. Having
this rationale in mind suggests that this mean-field mobility makes more sense
for a finite number $N$ of dumbbells, than it does for infinitely many. In the
case of the potential energy, the microscopic physics of the implied coupling
between dumbbells is less clear. Still, we find it worthy of note that a
potential energy function exists from which the FENE-P model can be derived
exactly.

In order to highlight the overall structure of the modeling in the remainder
of this paper, the general forms eq. (6) for the potential $\Phi$ and eq. (9)
for the mobility tensor $\bm{M}$ will be used. These general results can then
be reduced to the Hookean dumbbell model, the FENE-P model, and the Giesekus
model, respectively, by appropriate choices for the forms and parameters for
the potential $\Phi$ and the mobility tensor $\bm{M}$, see table 1.

Table 1: Overview of the parameters in the potential $\Phi$ given by eq. (7) and eq. (8), and the mobility tensor $\bm{M}$, eq. (10), for the three models Dumbbell | Parameters:
---|---
model | $\Phi$ | $\bm{M}$
Hookean |  $\beta_{1}=1$, $\beta_{2}\to\infty$ ($f=1$) | $\alpha=0$
FENE-P |  $\beta_{1}=1$, $\beta_{2}$ finite ($f\neq 1$) | $\alpha=0$
Giesekus |  $\beta_{1}=1$, $\beta_{2}\to\infty$ ($f=1$) | $\alpha>0$

###  2.2 Transition from dumbbells to the conformation tensor

Given the definition of the instantaneous conformation tensor $\hat{\bm{c}}$,
eq. (5), and using the Itô interpretation of stochastic calculus McKean ;
hcostochbook , one has in general

$\displaystyle{\rm{d}}\hat{\bm{c}}=\frac{1}{N}\sum^{N}_{\mu=1}\left(({\rm{d}}\bm{Q}_{\mu})\bm{Q}_{\mu}+\bm{Q}_{\mu}({\rm{d}}\bm{Q}_{\mu})+\left\langle({\rm{d}}\bm{Q}_{\mu})({\rm{d}}\bm{Q}_{\mu})\right\rangle^{\textrm{It{\^{o}}}}_{{\rm{d}}t}\right)\,,$
(11)

where $\left\langle\ldots\right\rangle^{\textrm{It{\^{o}}}}_{{\rm{d}}t}$
implies that in ${\rm{d}}\bm{Q}_{\mu}$ only terms involving the Wiener
increments are kept and subsequently reduced according to the rule (see Table
3.1 in hcostochbook )

$\displaystyle{\rm{d}}\bm{W}_{\mu}{\rm{d}}\bm{W}_{\mu}$
$\displaystyle\rightarrow$ $\displaystyle{\rm{d}}t\bm{1}\,.$ (12)

Applied to the above class of models, the SDEs (1) for the dumbbell vectors
$\\{\bm{Q}_{\mu}\\}_{\mu=1,\ldots,N}$ can be transformed into an SDE for the
conformation tensor,

$\displaystyle{\rm{d}}\hat{\bm{c}}$ $\displaystyle=$
$\displaystyle\left[\bm{\kappa}\cdot\hat{\bm{c}}+\hat{\bm{c}}\cdot\bm{\kappa}^{\rm{T}}-2\bm{M}\cdot\left(2\frac{\partial\Phi_{\rm
d}}{\partial({\rm{tr}\,}{\hat{\bm{c}}})}\hat{\bm{c}}-k_{\rm{B}}T\bm{1}\right)\right]{\rm{d}}t$
(13)
$\displaystyle+\frac{2k_{\rm{B}}T}{N}\left[\hat{\bm{c}}\cdot\frac{\partial}{\partial\hat{\bm{c}}}\cdot\bm{M}+\left(\hat{\bm{c}}\cdot\frac{\partial}{\partial\hat{\bm{c}}}\cdot\bm{M}\right)^{\rm{T}}\right]{\rm{d}}t+{\rm{d}}\hat{\bm{c}}^{\rm{f}}\,,$

in terms of the dumbbell potential energy $\Phi_{\rm d}$ and the dumbbell
mobility tensor $\bm{M}$, where it has been assumed that
$\bm{M}\cdot\hat{\bm{c}}=\hat{\bm{c}}\cdot\bm{M}$. It is pointed out that the
symmetry of $\hat{\bm{c}}$ must be taken into account explicitly when
calculating the partial derivatives of $\bm{M}$ (see HHA2018 for details). The
quantity ${\rm{d}}\hat{\bm{c}}^{\rm{f}}$ denotes the thermal fluctuations,

$\displaystyle{\rm{d}}\hat{\bm{c}}^{\rm{f}}=\frac{\sqrt{2k_{\rm{B}}T}}{N}\sum^{N}_{\mu=1}\left(\left(\bm{B}\cdot{\rm{d}}\bm{W}_{\mu}\right)\bm{Q}_{\mu}+\bm{Q}_{\mu}\left(\bm{B}\cdot{\rm{d}}\bm{W}_{\mu}\right)\right)\,,$
(14)

with $\bm{B}$ given by eq. (2) for a general mobility tensor $\bm{M}$, eq.
(9). It can be shown that the fluctuations have the properties

$\displaystyle\langle{\rm{d}}\hat{c}^{\rm{f}}_{ij}\rangle$ $\displaystyle=$
$\displaystyle 0\,,$ (15)
$\displaystyle\langle{\rm{d}}\hat{c}^{\rm{f}}_{ij}{\rm{d}}\hat{c}^{\rm{f}}_{kl}\rangle$
$\displaystyle=$
$\displaystyle\frac{2k_{\rm{B}}T}{N}\left(\hat{c}_{ik}M_{jl}+\hat{c}_{il}M_{jk}\right.$
(16)
$\displaystyle\qquad\quad\left.+\hat{c}_{jk}M_{il}+\hat{c}_{jl}M_{ik}\right){\rm{d}}t\,.$

The SDE (13) for the conformation tensor with fluctuations obeying the
statistical properties given by eq. (15) and eq. (16) is the benchmark to
which the thermodynamic treatment further below will be compared, for the
general class of models described by energy $\Phi$, eq. (6), and mobility
tensor $\bm{M}$, eq. (9). If the potential $\Phi$ and the mobility tensor
$\bm{M}$ are of forms more general than eq. (6) and eq. (9), respectively, the
dynamics for the conformation tensor would not close automatically, in which
case one would have to employ procedures of coarse graining hcoCG ; hcobook ;
espanol2002 ; espanol2009 . The procedure presented in this sec. 2.2 has also
been followed in vazquezquesadaetal2009 for deriving the
$\hat{\bm{c}}$-dynamics for the Hookean and FENE-P models.

It is noted that, in the limit of many dumbbells, i.e., $N\to\infty$, not only
do the fluctuations ${\rm{d}}\hat{\bm{c}}^{\rm{f}}$ become insignificant. In
addition, also the “thermal drift”, i.e., the second-last contribution on the
right-hand side (r.h.s.) of eq. (13) vanishes as well, and thus the
conventional deterministic dynamics for the conformation tensor is recovered.

The Hookean dumbbell model – which results in the Maxwell model for the
conformation tensor – , the FENE-P model, and the Giesekus model are sub-cases
of the SDE (13) with noise (14) when choosing the parameters $\beta_{1}$,
$\beta_{2}$, and $\alpha$ appropriately, see table 1.

##  3 Fluctuating viscoelasticity derived using thermodynamics

The main idea in taking a thermodynamic approach to modeling dynamical systems
is that one can concentrate on key ingredients for the static and dynamic
properties, and the thermodynamic approach makes sure that these ingredients
are processed towards the final model in a consistent way. For the dumbbell
dynamics (1), the key ingredients are the potential $\Phi$ and the mobility
tensor $\bm{M}$. In contrast, for the dynamics of $\bm{c}$, the key
ingredients are the Helmholtz free energy density $\psi$ and the generalized
relaxation tensor $\bm{\Lambda}^{(4)}$ (see HHA2018 for details). In
particular, the dynamics for the fluctuating conformation-tensor $\bm{c}$ is
given by the SDE

$\displaystyle{\rm{d}}\bm{c}$ $\displaystyle=$
$\displaystyle\left(\bm{\kappa}\cdot\bm{c}+\bm{c}\cdot\bm{\kappa}^{{\rm{T}}}-\bm{\Lambda}^{(4)}\odot\frac{\partial\psi}{\partial\bm{c}}\right){\rm{d}}t$
(17) $\displaystyle+\frac{k_{\rm{B}}T}{V}{\rm
div}_{\bm{c}}\bm{\Lambda}^{(4)}{\rm{d}}t+\sqrt{\frac{2k_{\rm{B}}T}{V}}\bm{B}^{(4)}\odot{\rm{d}}\tilde{\bm{W}}\,,$

where ${\rm div}_{\bm{c}}\bm{\Lambda}^{(4)}$ denotes the divergence of
$\bm{\Lambda}^{(4)}$ in $\bm{c}$-space, the order-four tensor $\bm{B}^{(4)}$
satisfies

$\displaystyle\bm{B}^{(4)}\odot\bm{B}^{(4),{\rm{T}}}=\bm{\Lambda}^{(4)}\,,$
(18)

and ${\rm{d}}\tilde{\bm{W}}$ is a tensor with increments of independent Wiener
processes (see HHA2018 for further details). The structure of the SDE (17)
with eq. (18) is completely analogous to the one for the dumbbell models, eq.
(1) with eq. (2).

In the following, we derive expressions for $\psi$ and $\bm{\Lambda}^{(4)}$,
based on those for $\Phi$ and $\bm{M}$, respectively.

###  3.1 Free energy density $\psi(\bm{c})$ for finite $N$

The Helmholtz free energy $\Psi=\Psi(\bm{c})$ for the symmetric conformation-
tensor $\bm{c}$ for a finite number $N$ of dumbbells is given by
$\Psi=-k_{\rm{B}}T\ln Z$, with the canonical partition-function

$\displaystyle Z(\bm{c})=\int
e^{-\Phi(\hat{\bm{c}})/(k_{\rm{B}}T)}\,\delta^{(K)}\left(\hat{\bm{c}}-\bm{c}\right)\,d^{DN}Q\,,$
(19)

where $D$ is the number of spatial dimensions. The $K$-dimensional Dirac
$\delta$-function makes sure that only those states in
$\\{\bm{Q}_{\mu}\\}$-space are accounted for that are compatible with the
conformation tensor $\bm{c}$. Since $\hat{\bm{c}}$ is symmetric by definition,
see eq. (5), only $K=D(D+1)/2$ independent conditions are needed (instead of
$D^{2}$); no more conditions are required for properly restricting the
integration in $\\{\bm{Q}_{\mu}\\}$-space.

It is pointed out that $\delta^{(K)}$ is actually a $\delta$-function in
$\bm{c}$-space in the sense that
$\int\delta^{(K)}(\hat{\bm{c}}-\bm{c})d^{K}c=1$. Using this latter relation,
the integral of $Z(\bm{c})$ over all (symmetric) conformation tensors $\bm{c}$
reduces to

$\displaystyle\Xi$ $\displaystyle=$ $\displaystyle\int Z(\bm{c})d^{K}c=\int
e^{-\Phi(\hat{\bm{c}})/(k_{\rm{B}}T)}\,d^{DN}Q\,,$ (20)

which is the conventional canonical partition function in the absence of the
constraint $\hat{\bm{c}}=\bm{c}$.

As we restrict our attention to energy functions $\Phi$ which depend on
$\\{\bm{Q}_{\mu}\\}_{\mu=1,\ldots,N}$ only by way of $\hat{\bm{c}}$, see eq.
(6), the canonical partition-function $Z$ is related to the microcanonical
partition-function $\Gamma$ by way of

$\displaystyle Z(\bm{c})=e^{-\Phi(\bm{c})/(k_{\rm{B}}T)}\,\Gamma(\bm{c})\,,$
(21)

with

$\displaystyle\Gamma(\bm{c})=\int\,\delta^{(K)}\left(\hat{\bm{c}}-\bm{c}\right)\,d^{DN}Q\,.$
(22)

Different procedures for calculating the dependence of $\Gamma$ on $\bm{c}$
explicitly are discussed in Appendix B, one based on deriving a differential
equation for $\Gamma$, another one with a more geometrical interpretation, and
a third one using a scaling argument. Following any of these procedures, the
result for finite $N$ is

$\displaystyle\Gamma=\Gamma_{0}\left(\det\bm{c}\right)^{(N-D-1)/2}\,,$ (23)

with a $\bm{c}$-independent prefactor $\Gamma_{0}$.

Based on eq. (21) with eq. (23), the Helmholtz free energy density
$\psi=\Psi/V$ per volume $V$ becomes

$\displaystyle\psi=\frac{\Phi}{V}-\frac{nk_{\rm{B}}T}{2}\ln\left(\det\bm{c}\right)+\Delta\psi\,,$
(24)

with

$\displaystyle\Delta\psi=\frac{k_{\rm{B}}T}{2V}(D+1)\ln\left(\det\bm{c}\right)\,,$
(25)

and number density $n=N/V$, and where we have omitted a $\bm{c}$-independent
additive constant, which is irrelevant for the formulation of the dynamics of
$\bm{c}$ according to eq. (17). Since $\Phi$ is proportional to $N$, it is
evident that the first two contributions on the r.h.s. of eq. (24) are
independent of the size of the system, for given number density $n$. In
contrast, the third contribution, $\Delta\psi$, does depend on the size of the
system, in particular it becomes more relevant the smaller the system. To the
best of our knowledge, this finite-size correction to the Helmholtz free
energy (density) has not been derived earlier.

Using $H=k_{\rm{B}}T$ and $G=nk_{\rm{B}}T$, the Helmholtz free energy density
(24) with eq. (25) for the three models discussed in table 1 agrees with
standard literature (e.g., see birdbook ; berisedwards1990b ;
wapperomhulsen1998 ), and with what has been used in the fluctuating-
viscoelasticity approach in HHA2018 , with the important difference of the
finite-size correction $\Delta\psi$. Using, in contrast to our procedure, a
continuous (representative of $N\to\infty$) distribution for the dumbbell
vector $\bm{Q}$ (e.g., see birdbook ), the thereby-derived Helmholtz free
energy density corresponds to eq. (24) where the finite-size correction
$\Delta\psi$ is absent.

###  3.2 Relaxation tensor $\bm{\Lambda}^{(4)}$

In the dumbbell dynamics (1), structural relaxation is expressed as
$-\bm{M}\cdot\left(\partial\Phi/\partial\bm{Q}_{\mu}\right){\rm{d}}t$. In the
$\bm{c}_{\mu}$-dynamics (17), structural relaxation is expressed as
$-\bm{\Lambda}^{(4)}\odot\left(\partial\psi/\partial\bm{c}\right){\rm{d}}t$,
with an order-four relaxation tensor $\bm{\Lambda}^{(4)}$ HHA2018 . In the
translation from $\bm{M}$ to $\bm{\Lambda}^{(4)}$, a reduction of variables
and the volume of the system $V$ are involved, the latter being necessary
since $\Phi$ is an energy while $\psi$ is an energy density. The relation
between $\bm{M}$ and $\bm{\Lambda}^{(4)}$ is given by (see also hcoCG , and
sect. 6.4 in hcobook )

$\displaystyle\bm{\Lambda}^{(4)}=V\left\langle\sum^{N}_{\mu=1}\frac{\partial\hat{\bm{c}}}{\partial\bm{Q}_{\mu}}\cdot\bm{M}\cdot\frac{\partial\hat{\bm{c}}}{\partial\bm{Q}_{\mu}}\right\rangle\,,$
(26)

where the contractions run over the components of $\bm{Q}_{\mu}$ and $\bm{M}$,
and $\langle\ldots\rangle$ is the average over $\\{\bm{Q}_{\mu}\\}$-space for
given $\bm{c}$. Using $\partial\hat{c}_{ij}/\partial
Q_{\mu,m}=(\delta_{im}Q_{\mu,j}+\delta_{jm}Q_{\mu,i})/N$, one finds

$\displaystyle\Lambda^{(4)}_{ijkl}=\frac{V}{N}\left\langle\hat{c}_{jl}M_{ik}+\hat{c}_{jk}M_{il}+\hat{c}_{il}M_{jk}+\hat{c}_{ik}M_{jl}\right\rangle\,.$
(27)

Since $\bm{M}$ depends on the positions $\\{\bm{Q}_{\mu}\\}_{\mu=1,\ldots,N}$
only by way of $\hat{\bm{c}}$, taking the average is thus equivalent to
replacing $\hat{\bm{c}}$ by $\bm{c}$ everywhere in eq. (27). It is to be noted
that there is no finite-size correction in $\bm{\Lambda}^{(4)}$.

Using again $G=nk_{\rm{B}}T$ and with $\zeta=4k_{\rm{B}}T\lambda$, the
relaxation tensor $\bm{\Lambda}^{(4)}$ given by eq. (27) for the three models
in table 1 turns out to agree with the standard expressions in the literature
(e.g., see birdbook ; berisedwards1990b ; wapperomhulsen1998 ) , and with what
has been used in HHA2018 in the context of fluctuating viscoelasticity.

###  3.3 Application to fluctuating viscoelasticity

According to the general procedure in HHA2018 , represented in eq. (17), and
using the Helmholtz free energy density (24) with eq. (25) and the relaxation
tensor (27) with $\hat{\bm{c}}\to\bm{c}$, one observes that the results in
HHA2018 need to be amended by including the finite-$N$ contribution

$\displaystyle\left.{\rm{d}}\bm{c}\right|_{\Delta\psi}$ $\displaystyle=$
$\displaystyle-\bm{\Lambda}^{(4)}\odot\frac{\partial(\Delta\psi)}{\partial\bm{c}}{\rm{d}}t$
(28a) $\displaystyle=$ $\displaystyle-2k_{\rm{B}}T((D+1)/N)\bm{M}{\rm{d}}t\,.$
(28b)

In particular, one obtains for the complete fluctuating dynamics

$\displaystyle{\rm{d}}\bm{c}$ $\displaystyle=$
$\displaystyle\left[\bm{\kappa}\cdot\bm{c}+\bm{c}\cdot\bm{\kappa}^{{\rm{T}}}-2\bm{M}\cdot\left(2\frac{\partial\Phi_{\rm
d}}{\partial({\rm{tr}\,}{\bm{c}})}\bm{c}-k_{\rm{B}}T\bm{1}\right)\right]{\rm{d}}t$
(29)
$\displaystyle+\frac{2k_{\rm{B}}T}{N}\left[\hat{\bm{c}}\cdot\frac{\partial}{\partial\hat{\bm{c}}}\cdot\bm{M}+\left(\hat{\bm{c}}\cdot\frac{\partial}{\partial\hat{\bm{c}}}\cdot\bm{M}\right)^{\rm{T}}\right]{\rm{d}}t+{\rm{d}}\bm{c}^{\rm{f}}\,,$

where the symmetry of $\bm{c}$ has been taken into account when calculating
${\rm div}_{\bm{c}}\bm{\Lambda}^{(4)}$ (see HHA2018 for details). See Appendix
A for explicit exemplary applications of this equation. The symbol
${\rm{d}}\bm{c}^{\rm{f}}$ denotes the fluctuating contribution given by

$\displaystyle{\rm{d}}\bm{c}^{\rm{f}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2k_{\rm{B}}T}{N}}\left(\bm{b}\cdot{\rm{d}}\tilde{\bm{W}}\cdot\bm{B}^{\rm{T}}+\bm{B}\cdot{\rm{d}}\tilde{\bm{W}}^{\rm{T}}\cdot\bm{b}^{\rm{T}}\right)\,,$
(30)

where $\bm{b}$ satisfies the condition

$\displaystyle\bm{c}=\bm{b}\cdot\bm{b}^{\rm{T}}\,.$ (31)

In general, $\bm{B}$ in eq. (30) has dimensions $D\times P$ with $P\geq D$ (as
described in sect. 2.1), $\bm{b}$ has dimensions $D\times P^{\prime}$ with
$P^{\prime}\geq D$, and therefore ${\rm{d}}\tilde{\bm{W}}$ has dimensions
$P^{\prime}\times P$; for practical purposes, one may choose $P=P^{\prime}=D$.
The tensor ${\rm{d}}\tilde{\bm{W}}$ consists of increments of statistically
independent Wiener processes, with the properties

$\displaystyle\langle{\rm{d}}\tilde{W}_{ij}(t)\rangle$ $\displaystyle=$
$\displaystyle 0\,,$ (32)
$\displaystyle\langle{\rm{d}}\tilde{W}_{ij}(t)\,{\rm{d}}\tilde{W}_{kl}(t^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\delta_{ik}\,\delta_{jl}\,\delta(t-t^{\prime})\,dt\,dt^{\prime}\,.$
(33)

When using component notation, eq. (33) implies that any two of the components
of ${\rm{d}}\tilde{\bm{W}}$ are independent from each other.

A direct comparison shows that the SDE (13) derived directly from the dumbbell
model and the SDE (29) derived via the thermodynamic route, respectively,
agree. It is noted that the expressions for the fluctuations,
${\rm{d}}\hat{\bm{c}}^{\rm{f}}$ in eq. (14) and ${\rm{d}}\bm{c}^{\rm{f}}$ in
eq. (30), respectively, have a different form. However, it can be shown that
they have the same statistical properties. First, both representations are
linear superpositions of increments of Wiener processes, and second, for both
representations the average is given by eq. (15) and the covariance by (16).
The difference in the expressions for the fluctuations is not a short-coming
of the approach; it rather reflects the non-uniqueness of the decompositions
(18) and (31). The non-uniqueness of the decomposition (31) can actually be
utilized for relating the expressions (14) and (30) for the noise in even more
explicit terms. Specifically, choosing $\bm{b}$ to be $D\times N$ with the
column vectors of $\bm{b}$ equal to $\bm{Q}_{\mu}/\sqrt{N}$ ($\mu=1,\ldots,N$)
(see sect. 4 for a further elaboration), and setting the row vectors of
${\rm{d}}\tilde{\bm{W}}$ equal to ${\rm{d}}\bm{W}_{\mu}$ ($\mu=1,\ldots,N$),
one finds that the expressions (14) and (30) are identical.

In deriving the relaxation term in eq. (29), i.e. the first term on the r.h.s.
that is proportional to $\bm{M}$, from the general form eq. (17), one notices
the following: The prefactor $1/N$ in $\bm{\Lambda}^{(4)}$ given by eq. (27)
is cancelled by the prefactor $N$ in the derivative
$\partial(\psi-\Delta\psi)/\partial\bm{c}$, for $\psi$ given by eq. (24) with
$\Phi$ according to eq. (6). If one chose to not cancel these factors, one
would observe that $(1/N)\bm{M}$ (with factor $1/N$) is the relevant mobility
on the conformation-tensor level not only for the thermal drift and the
fluctuations (by way of the covariances (16)), but also for the relaxation.

The importance of the finite-size correction $\Delta\psi$ in the free energy
density, eq. (25), for the evolution equation (29) is pointed out. In
particular, $\left.{\rm{d}}\bm{c}\right|_{\Delta\psi}$ exactly cancels those
contributions from ${\rm div}_{\bm{c}}\bm{\Lambda}^{(4)}$ that are related to
the derivative of the explicit factors $\hat{\bm{c}}$ in $\bm{\Lambda}^{(4)}$,
eq. (27). If $\Delta\psi$ was neglected, agreement between the SDEs (13) and
(29) could not be achieved.

In Appendix A, the dynamics for the conformation tensor with fluctuations, eq.
(29) with eq. (30), is presented explicitly for three models, namely for the
Hookean dumbbell / Maxwell model, the FENE-P model, and the Giesekus model.

##  4 Comments on the multiplicative decomposition of c

###  4.1 Eliminating degrees of freedom

Above, the relation has been established between the dynamics formulated in
terms of dumbbell vectors, on the one hand, and in terms of the conformation
tensor, on the other hand. In this section, the relation of the dumbbell-
vector description to the multiplicative decomposition of the conformation
tensor (e.g., see HHA2018 ; HCHA2020 ),

$\displaystyle\hat{\bm{c}}=\hat{\bm{b}}_{P^{\prime}}\cdot\hat{\bm{b}}_{P^{\prime}}^{\rm{T}}\,,$
(34)

is examined, where $\hat{\bm{b}}$ has dimensions $D\times P^{\prime}$ with
$P^{\prime}$ an arbitrary dimension. This decomposition can be written in the
form

$\displaystyle\hat{\bm{c}}=\sum_{\nu=1}^{P^{\prime}}\hat{\bm{b}}_{P^{\prime},\nu}\hat{\bm{b}}_{P^{\prime},\nu}\,,$
(35)

with $\hat{\bm{b}}_{P^{\prime},\nu}$ the $\nu$-th column vector of
$\hat{\bm{b}}_{P^{\prime}}$ (see also publ72 ). In general, one obviously must
require $P^{\prime}\geq D$ for this decomposition to be complete for arbitrary
conformation tensor $\hat{\bm{c}}$.

In view of the expression (5) for the conformation tensor, a natural choice is
$P^{\prime}=N$ with $\hat{\bm{b}}_{N,\mu}=\bm{Q}_{\mu}/\sqrt{N}$, relating the
dynamics of $\hat{\bm{b}}_{N}$ directly to that of the dumbbell vectors
$\bm{Q}_{\mu}$. In the following, we focus on Hookean dumbbells, i.e., the
Maxwell model, for illustrative purposes. The evolution equations (1) for
Hookean dumbbells, using the potential (8) and mobility tensor (10) with the
parameters given in table 1, become

$\displaystyle{\rm{d}}\bm{Q}_{\mu}=\left[\bm{\kappa}\cdot\bm{Q}_{\mu}-\frac{1}{2\lambda}\bm{Q}_{\mu}\right]{\rm{d}}t+\frac{1}{\sqrt{\lambda}}{\rm{d}}\bm{W}_{\mu}\,,$
(36)

where the identifications $\lambda=\zeta/(4H)$ and $H=k_{\rm{B}}T$ have been
made. Eq. (36) translates directly into the dynamics of $\hat{\bm{b}}_{N}$,

$\displaystyle{\rm{d}}\hat{\bm{b}}_{N}=\left[\bm{\kappa}\cdot\hat{\bm{b}}_{N}-\frac{1}{2\lambda}\hat{\bm{b}}_{N}\right]{\rm{d}}t+\frac{1}{\sqrt{N\lambda}}{\rm{d}}\check{\bm{W}}_{N}\,,$
(37)

where ${\rm{d}}\check{\bm{W}}_{N}$ has dimensions $D\times N$ with its
$\mu$-th column vector given by ${\rm{d}}\bm{W}_{\mu}$. Let us now compare
this result with the dynamics for the “square root” $\bm{b}_{D}$ of the
conformation tensor $\bm{c}$, i.e. $\bm{c}=\bm{b}_{D}\cdot\bm{b}_{D}^{\rm{T}}$
where $\bm{b}_{D}$ has dimensions $D\times D$, as derived in HHA2018 and
amended in HCHA2020 ,

$\displaystyle{\rm{d}}\bm{b}_{D}$ $\displaystyle=$
$\displaystyle\left[\bm{\kappa}\cdot\bm{b}_{D}-\frac{1}{2\lambda}(\bm{b}_{D}-\bm{b}^{-1,{\rm{T}}}_{D})-\frac{D}{2N\lambda}\bm{b}^{-1,{\rm{T}}}_{D}\right]{\rm{d}}t$
(38) $\displaystyle+\frac{1}{\sqrt{N\lambda}}{\rm{d}}\check{\bm{W}}_{D}\,.$

The close relation between $\bm{b}_{D}$ and the elastic, i.e., recoverable,
part of the deformation gradient in solid mechanics has been discussed in
HHA2018 . For $N\to\infty$, the dynamics of the column vectors of $\bm{b}_{3}$
agree with the treatment proposed in Cho2009 .

Two major differences between eq. (37) and eq. (38) are apparent. First, the
relaxation in eq. (37) drives the column vectors of $\hat{\bm{b}}_{N}$ to
zero, while , considering e.g. $D=3$, according to eq. (38) $\bm{b}_{3,1}$
relaxes to $\bm{b}_{3,2}\times\bm{b}_{3,3}/\det{\bm{b}_{3}}$, $\bm{b}_{3,2}$
to $\bm{b}_{3,3}\times\bm{b}_{3,1}/\det{\bm{b}_{3}}$, and $\bm{b}_{3,3}$ to
$\bm{b}_{3,1}\times\bm{b}_{3,2}/\det{\bm{b}_{3}}$, respectively, i.e., the
column vectors of $\bm{b}_{3}$ become orthonormal in the course of relaxation.
The second major difference between eq. (37) and eq. (38) relates to the
absence of the thermal drift (the third term on the r.h.s. of eq. (38)) in eq.
(37). Both differences, in relaxation and thermal drift, are a hallmark of
eliminating degrees of freedom when going to a reduced description of the
dynamics. More specifically, both of these contributions are tightly related
to the entropy, i.e., to the counting of states in configuration space, see
sect. 3.1. It is pointed out that the difference in relaxation does not depend
on the value of $N$, while the thermal drift clearly is a finite-$N$ effect,
i.e., it is related to the fluctuations. However, despite these differences
between eq. (37) and eq. (38), one should keep in mind that they both result
in the same dynamics for the conformation tensor.

###  4.2 Rotational dynamics

In HCHA2020 , it has been discussed that, due to the non-uniqueness of the
decomposition $\bm{c}=\bm{b}_{D}\cdot\bm{b}_{D}^{\rm{T}}$, there is on-going
rotational dynamics in eq. (38) even at equilibrium, with a relaxation time
that depends approximately linearly on $N$. In the following, an attempt is
made to rationalize this $N$-dependence of the rotational relaxation time in
terms of the dynamics for the dumbbell vectors $\bm{Q}_{\mu}$, eq. (36).
Consider the linear combination

$\displaystyle\bm{R}=\sum^{N}_{\mu=1}\gamma_{\mu}\bm{Q}_{\mu}\,.$ (39)

Based on eq. (36), its dynamics is given by

$\displaystyle{\rm{d}}\bm{R}=\left[\bm{\kappa}\cdot\bm{R}-\frac{1}{2\lambda}\bm{R}\right]{\rm{d}}t+\frac{1}{\sqrt{\lambda}}{\rm{d}}\bar{\bm{W}}\,,$
(40)

where the Wiener process increments defined by
${\rm{d}}\bar{\bm{W}}=\sum^{N}_{\mu=1}\gamma_{\mu}{\rm{d}}\bm{W}_{\mu}$
satisfy

$\displaystyle\langle{\rm{d}}\bar{\bm{W}}\rangle$ $\displaystyle=$
$\displaystyle\bm{0}\,,$ (41)
$\displaystyle\langle{\rm{d}}\bar{\bm{W}}{\rm{d}}\bar{\bm{W}}\rangle$
$\displaystyle=$ $\displaystyle\overline{\gamma^{2}}{\rm{d}}t\bm{1}\,,$ (42)

with $\overline{\gamma^{2}}\equiv\sum^{N}_{\mu=1}\gamma_{\mu}^{2}$. Let us
consider equilibrium, $\bm{\kappa}=\bm{0}$. In order to study the orientation
dynamics, we write $\bm{R}=R\bm{n}$, with $R$ and $\bm{n}$ the length and the
orientation vector of $\bm{R}$, respectively. By using Itô calculus
gardinerbook ; hcostochbook , it can be shown that the SDEs for $R$ and
$\bm{n}$ are given by

$\displaystyle{\rm{d}}R$ $\displaystyle=$
$\displaystyle\left(\frac{\overline{\gamma^{2}}}{2R\lambda}\left(D-1\right)-\frac{1}{2\lambda}R\right){\rm{d}}t+\frac{1}{\sqrt{\lambda}}\bm{n}\cdot{\rm{d}}\bar{\bm{W}}\,,$
(43) $\displaystyle{\rm{d}}\bm{n}$ $\displaystyle=$
$\displaystyle-\frac{\overline{\gamma^{2}}}{2R^{2}\lambda}\left(D-1\right)\bm{n}{\rm{d}}t+\frac{1}{R\sqrt{\lambda}}\left(\bm{1}-\bm{n}\bm{n}\right)\cdot{\rm{d}}\bar{\bm{W}}\,.$
(44)

The contributions proportional to $\overline{\gamma^{2}}$ originate from the
second-order term in the Itô calculus 222Note: It can be shown, again by using
Itô calculus gardinerbook ; hcostochbook , that indeed ${\rm{d}}|\bm{n}|=0$..
In particular, one observes in the orientation dynamics (44) that the
relaxation time for the orientation vector $\bm{n}$ is given by

$\displaystyle\lambda_{n}$ $\displaystyle=$
$\displaystyle\frac{2R^{2}}{\overline{\gamma^{2}}\left(D-1\right)}\lambda\,.$
(45)

As an example, consider the case $D=3$. To get an idea about the rotational
dynamics of the column vectors $\bm{b}_{3,\mu}$ of $\bm{b}_{3}$, which satisfy
$\bm{c}=\sum_{\mu=1}^{3}\bm{b}_{3,\mu}\bm{b}_{3,\mu}$, we make a particular
choice for $\bm{R}$, i.e., for the linear combination (39). To that end, think
of a Voronoi tessellation on the sphere, generated by the six “poles” that
themselves are generated as intersections of three orthogonal axes with the
sphere. Let us now consider two of these Voronoi sectors, $\mathcal{V}_{i}$
and $\mathcal{V}_{\bar{i}}$, which are on opposite sides of the sphere. It is
reasonable to assume that the number of vectors $\bm{Q}_{\mu}$ which have
their orientation in these two Voronoi sectors together can be written as
$\varphi N$, where $\varphi$ is independent of $N$. After mapping all vectors
$\bm{Q}_{\mu}$ with orientation in $\mathcal{V}_{\bar{i}}$ into
$\mathcal{V}_{i}$ by way of $\bm{Q}_{\mu}\to-\bm{Q}_{\mu}$, which is just
making use of the symmetry of dumbbell description, all $\varphi N$ vectors
with orientation in $\mathcal{V}_{i}$ are averaged to obtain $\bm{R}$, i.e.,
$\gamma_{\mu}=1/(\varphi N)$ for these vectors and $\gamma_{\mu}=0$ for all
others. In this case, one finds $\overline{\gamma^{2}}=1/(\varphi N)$, and
thus the relaxation time for rotation (45) is increased w.r.t. $\lambda$,
namely as $\lambda_{n}\propto N\lambda$, in agreement with the observation in
HCHA2020 .

For the construction of a suitable vector $\bm{R}$, the Voronoi sectors have
been used, instead of, e.g., considering a random selection of vectors
$\bm{Q}_{\mu}$, for two reasons: First, the construction with Voronoi sectors
allows to construct three such vectors (representative of the column vectors
of $\bm{b}_{3}$) that are clearly linearly independent from each other. And
second, the length of $\bm{R}$ is proportional to the length of the dumbbell
vectors $\bm{Q}_{\mu}$ with a prefactor that is of order $\mathcal{O}(1/2)$,
i.e. independent of $N$.

##  5 Discussion and conclusions

The focus of this paper has been on deriving viscoelasticity in terms of a
conformation tensor with fluctuations, based on the kinetic theory of
dumbbells. This has been achieved by identifying the conformation tensor with
the arithmetic average over a finite number $N$ of dumbbells, eq. (5), and
using two alternative routes for deriving the dynamics: a direct approach
using stochastic calculus, and a thermodynamic approach, in which the
Helmholtz free energy plays a key role. It has been shown that these two
approaches agree only if a finite-size contribution to the Helmholtz free
energy of the conformation tensor is taken into account. The main messages of
this paper are therefore the following:

  * •

If the number $N$ of dumbbells is finite, the commonly employed expressions
for the thermodynamic potentials need to be corrected: Using statistical
mechanics (see Appendix B), one finds that the conformational entropy must be
corrected by replacing $N$ by $N-D-1$ (with $D$ the number of spatial
dimensions), which in turn modifes the Helmholtz free energy, see eq. (24)
with correction term eq. (25).

  * •

The thereby obtained finite-size correction in the free energy is crucial for
guaranteeing compatibility between the dynamics of the conformation tensor and
of the underlying dumbbells.

While these general conclusions have been established in general terms for a
large class of models, they have also been exemplified for the Hookean
(Maxwell) model, the FENE-P model, and the Giesekus model (see Appendix A);
the dynamics for the conformation tensor with fluctuations for these three
models is summarized in table 2.

Table 2: Dynamics for the conformation tensor $\bm{c}$ with fluctuations; see Appendix A for the derivation. Symbols are explained in the text. Note that the square root of a tensor is symmetric. The increments ${\rm{d}}\tilde{\bm{W}}$ of Wiener processes satisfy $\langle{\rm{d}}\tilde{W}_{ij}(t)\rangle=0$ and $\langle{\rm{d}}\tilde{W}_{ij}(t)\,{\rm{d}}\tilde{W}_{kl}(t^{\prime})\rangle=\delta_{ik}\,\delta_{jl}\,\delta(t-t^{\prime})\,dt\,dt^{\prime}$ (see eq. (32) and eq. (33)).  Model | Conformation dynamics | Noise
---|---|---
Maxwell | $\displaystyle{\rm{d}}\bm{c}=\left[\bm{\kappa}\cdot\bm{c}+\bm{c}\cdot\bm{\kappa}^{\rm{T}}-\frac{4H}{\zeta}\left(\bm{c}-\frac{k_{\rm{B}}T}{H}\bm{1}\right)\right]{\rm{d}}t+{\rm{d}}\bm{c}^{\rm{f}}$ | $\displaystyle{\rm{d}}\bm{c}^{\rm{f}}=\sqrt{\frac{4k_{\rm{B}}T}{N\zeta}}\left(\sqrt{\bm{c}}\cdot{\rm{d}}\tilde{\bm{W}}+{\rm{d}}\tilde{\bm{W}}^{\rm{T}}\cdot\sqrt{\bm{c}}\right)$
FENE-P | $\displaystyle{\rm{d}}\bm{c}=\left[\bm{\kappa}\cdot\bm{c}+\bm{c}\cdot\bm{\kappa}^{\rm{T}}-\frac{4H}{\zeta}\left(f\,\bm{c}-\frac{k_{\rm{B}}T}{H}\bm{1}\right)\right]{\rm{d}}t+{\rm{d}}\bm{c}^{\rm{f}}$ | $\displaystyle{\rm{d}}\bm{c}^{\rm{f}}=\sqrt{\frac{4k_{\rm{B}}T}{N\zeta}}\left(\sqrt{\bm{c}}\cdot{\rm{d}}\tilde{\bm{W}}+{\rm{d}}\tilde{\bm{W}}^{\rm{T}}\cdot\sqrt{\bm{c}}\right)$
Giesekus | $\displaystyle{\rm{d}}\bm{c}=\left[\bm{\kappa}\cdot\bm{c}+\bm{c}\cdot\bm{\kappa}^{\rm{T}}-\frac{4H}{\zeta}\left((1-\alpha)\bm{1}+\alpha\bm{c}\right)\cdot\left(\bm{c}-\frac{k_{\rm{B}}T}{H}\bm{1}\right)\right]{\rm{d}}t$ | $\displaystyle\displaystyle\displaystyle{\rm{d}}\bm{c}^{\rm{f}}=\sqrt{\frac{4k_{\rm{B}}T}{N\zeta}}\left(\sqrt{\bm{c}}\cdot{\rm{d}}\tilde{\bm{W}}\cdot\bar{\bm{B}}+\bar{\bm{B}}\cdot{\rm{d}}\tilde{\bm{W}}^{\rm{T}}\cdot\sqrt{\bm{c}}\right)$
| $\qquad\displaystyle+\frac{4\alpha(D+1)k_{\rm{B}}T}{N\zeta}\bm{c}{\rm{d}}t+{\rm{d}}\bm{c}^{\rm{f}}$ |  where $\displaystyle\bar{\bm{B}}=\sqrt{(1-\alpha)\bm{1}+\alpha\bm{c}}$

When discussing a model with fluctuations, the deterministic counterpart
serves as a benchmark. For all models discussed in this paper, one recovers
the known deterministic models in the thermodynamic limit $N\to\infty$, i.e.,
if the number of dumbbells in the volume of interest $V$ diverges, keeping the
number density $n=N/V$ constant. Beyond that thermodynamic limit, however,
there is also an interest in the behavior of the average conformation tensor
$\langle\hat{\bm{c}}\rangle$ for finite $N$, i.e., in the presence of
fluctuations. For most models studied in this paper, the nonlinearities do not
allow to obtain a closed form equation for $\langle\hat{\bm{c}}\rangle$ based
on the stochastic differential equation (SDE), eq. (13) and eq. (29), for the
fluctuating conformation tensor $\hat{\bm{c}}$. The notable exception to this
rule is the Hookean dumbbell (Maxwell) model, with potential energy $\Phi_{\rm
d}({\rm tr}\left(\bm{Q}\bm{Q}\right))=(H/2){\rm tr}\left(\bm{Q}\bm{Q}\right)$
and mobility tensor $\bm{M}=(2/\zeta)\bm{1}$. Taking the average of the SDE
(A.1) over different realizations of the fluctuations, one observes that the
average conformation tensor $\langle\hat{\bm{c}}\rangle$ obeys the same
differential equation as its deterministic ($N\to\infty$) counterpart – an
observation that generally does not hold for nonlinear models. In particular,
one finds for the Hookean dumbbell model at equilibrium
$\langle\hat{\bm{c}}\rangle_{\textrm{eq}}=\bm{1}$, where $H=k_{\rm{B}}T$ has
been used. As a word of caution, it is pointed out that the conformation
tensor $\hat{\bm{c}}$ that minimizes the Helmholtz free energy density, eq.
(24) with eq. (25), is given by
$\hat{\bm{c}}_{\textrm{min}}=(1-(D+1)/N)\bm{1}$, i.e., it does depend on the
finite size ($N$) of the system. The fact that
$\hat{\bm{c}}_{\textrm{min}}\neq\langle\hat{\bm{c}}\rangle_{\textrm{eq}}$ is
not a contradiction; it merely points out that the distribution of thermal
fluctuations around the minimum is not symmetric.

The relevance of the finite-size correction of the Helmholtz free energy,
$\Delta\psi$ given by eq. (25), has been discussed primarily in the context of
formulating dynamics with fluctuations for the conformation tensor $\bm{c}$.
However, it is also of immediate consequence for the formulation of
fluctuating viscoelasticity in terms of the “square root” $\bm{b}_{3}$, where
$\bm{c}=\bm{b}_{3}\cdot\bm{b}_{3}^{\rm{T}}$. In HHA2018 , a thermodynamic
approach has been taken towards deriving the dynamics of $\bm{b}_{3}$, based
on the dynamics of $\bm{c}$. Therefore, if the thermodynamic potential for the
$\bm{c}$-dynamics contains a finite-size contribution, the same holds true
also for the thermodynamic potential for the $\bm{b}_{3}$-dynamics, see sect.
4.3 in HHA2018 for details. Beyond these implications for the thermodynamics
of a $\bm{b}_{3}$-formulation, the kinetic models for $N$ dumbbells have also
been employed in this paper to give an explanation for the existence of a
rotational relaxation time proportional to $N$ in the fluctuating dynamics of
$\bm{b}_{3}$, which has been observed earlier HCHA2020 .

###### Acknowledgements.

MH acknowledges stimulating discussions with Hans Christian Öttinger,
particularly in relation to the interpretation of eq. (38). PDO is grateful to
Georgetown University and the Ives Foundation for support. A preliminary
version of the derivation in Appendix B.2, for the $N\to\infty$ limit, was
developed by DJR during previous discussions with Joseph Peterson and Gary
Leal.

## Author contribution statement

Peter Olmsted initiated this collaboration of the authors, and Daniel Read
spotted the inconsistency between the results in HHA2018 and a direct kinetic-
theory approach. Markus Hütter has made the main contributions to the main
part of the manuscript, as well as to Appendices A, B.1 and B.3.1, while
Daniel Read and Peter Olmsted have developed Appendices B.2 and B.3.2,
respectively. All authors have discussed in detail about all parts of the
paper, and helped to bring the paper in its final form. All the authors have
read and approved the final manuscript.

## Conflict of interest

The authors declare that they have no conflict of interest.

##  Appendix A Dynamics of conformation tensor for three exemplary models

For completeness, three concrete realizations of the general dynamics for the
conformation tensor, eq. (29) with eq. (30), are provided in this section.

###  A.1 Hookean, i.e., Maxwell, model

Using the potential $\Phi$ in eq. (8) with eq. (7) in the limit $\beta_{1}=1$
and $\beta_{2}\to\infty$, and with the mobility tensor $\bm{M}$ given by eq.
(10) for $\alpha=0$ for Hookean dumbbells, the general equation (29) turns
into the Maxwell model with fluctuations (see also HHA2018 ; HCHA2020 ),

$\displaystyle{\rm{d}}\hat{\bm{c}}$ $\displaystyle=$
$\displaystyle\left[\bm{\kappa}\cdot\hat{\bm{c}}+\hat{\bm{c}}\cdot\bm{\kappa}^{\rm{T}}-\frac{4H}{\zeta}\left(\hat{\bm{c}}-\frac{k_{\rm{B}}T}{H}\bm{1}\right)\right]{\rm{d}}t+{\rm{d}}\hat{\bm{c}}^{\rm{f}}\,,$
(A.1)

where the fluctuations are determined by eq. (30) with eq. (2). In particular,
one may choose both $\bm{b}$ and $\bm{B}$ to have dimensions $D\times D$ and
to be symmetric, i.e., $\bm{b}=\sqrt{\bm{c}}$ and
$\bm{B}=\sqrt{\bm{M}}=\sqrt{2/\zeta}\,\bm{1}$. This result for the stochastic
dynamics of the conformation tensor is identical to what has been derived in
vazquezquesadaetal2009 .

###  A.2 FENE-P model

Using the potential $\Phi$ in eq. (8) with eq. (7), and with the mobility
tensor $\bm{M}$ given by eq. (10) for $\alpha=0$, the general equation (29)
turns into the FENE-P model with fluctuations (see also HHA2018 ; HCHA2020 ),

$\displaystyle{\rm{d}}\hat{\bm{c}}$ $\displaystyle=$
$\displaystyle\left[\bm{\kappa}\cdot\hat{\bm{c}}+\hat{\bm{c}}\cdot\bm{\kappa}^{\rm{T}}-\frac{4H}{\zeta}\left(f\hat{\bm{c}}-\frac{k_{\rm{B}}T}{H}\bm{1}\right)\right]{\rm{d}}t+{\rm{d}}\hat{\bm{c}}^{\rm{f}}\,.$
(A.2)

The fluctuations are determined by eq. (30) and eq. (2) where, as for the
Maxwell model, one may choose both $\bm{b}$ and $\bm{B}$ to have dimensions
$D\times D$ and to be symmetric, i.e., $\bm{b}=\sqrt{\bm{c}}$ and
$\bm{B}=\sqrt{\bm{M}}=\sqrt{2/\zeta}\,\bm{1}$.

###  A.3 Giesekus model

Using the potential $\Phi$ in eq. (8) with eq. (7) in the limit $\beta_{1}=1$
and $\beta_{2}\to\infty$, and with the mobility tensor $\bm{M}$ given by eq.
(10), the general equation (29) turns into the Giesekus model with
fluctuations (see also HHA2018 ; HCHA2020 ),

$\displaystyle{\rm{d}}\hat{\bm{c}}$ $\displaystyle=$
$\displaystyle\left[\bm{\kappa}\cdot\hat{\bm{c}}+\hat{\bm{c}}\cdot\bm{\kappa}^{\rm{T}}-\frac{4H}{\zeta}\left((1-\alpha)\bm{1}+\alpha\hat{\bm{c}}\right)\cdot\left(\hat{\bm{c}}-\frac{k_{\rm{B}}T}{H}\bm{1}\right)\right]{\rm{d}}t$
(A.3)
$\displaystyle+\frac{4\alpha(D+1)k_{\rm{B}}T}{N\zeta}\hat{\bm{c}}{\rm{d}}t+{\rm{d}}\hat{\bm{c}}^{\rm{f}}\,,$

where the fluctuations are determined by eq. (30) with eq. (2). One may choose
both $\bm{b}$ and $\bm{B}$ to have dimensions $D\times D$ and to be symmetric,
i.e., $\bm{b}=\sqrt{\bm{c}}$ and $\bm{B}=\sqrt{\bm{M}}$, where $\bm{M}$ is
given by eq. (10) for $\alpha\neq 0$. The thereby-obtained expression for the
fluctuations differs from that used in HHA2018 , while sharing the same
statistical properties. The difference in the expressions for the fluctuations
reflects the non-uniqueness of the decomposition (18).

Of the three models discussed explicitly in this Appendix A, the Giesekus
model is the only one for which there is a thermal drift in the dynamics of
$\hat{\bm{c}}$, i.e., the second-last contribution on the r.h.s. of eq. (A.3).
It is present only if the mobility of the dumbbells is anisotropic,
$\alpha\neq 0$. Beyond the $N$-dependence of the noise
${\rm{d}}\hat{\bm{c}}^{\rm{f}}$, the thermal drift is the second explicit
consequence of the finite size of the system.

##  Appendix B Calculation of partition function $\Gamma$

In this Appendix, three procedures are presented for calculating the
microcanonical partition function $\Gamma$, eq. (22), for a finite number $N$
of dumbbells.

###  B.1 Procedure 1: Differential equation

We start by noting that the microcanonical partition function $\Gamma$, eq.
(22), contains only $K=D(D+1)/2$ (rather than $D^{2}$) Dirac
$\delta$-functions, representative of the constraints related to the
independent components $c_{ij}$ with $1\leq i\leq j\leq D$. The following
notation is introduced:

$\displaystyle\Delta_{ij}$ $\displaystyle=$
$\displaystyle\delta\left(\hat{c}_{ij}-c_{ij}\right)\,,\qquad i\leq j\,,$
(B.1) $\displaystyle\Delta^{[K]}$ $\displaystyle=$
$\displaystyle\prod_{k,l=1;k\leq l}^{D}\Delta_{kl}\,,$ (B.2)
$\displaystyle\Delta^{[K-1]}_{ij}$ $\displaystyle=$
$\displaystyle\prod_{k,l=1;k\leq l;k\neq i;l\neq j}^{D}\Delta_{kl}\,,\qquad
i\leq j\,,$ (B.3)

so that $\Delta^{[K-1]}_{ij}\Delta_{ij}=\Delta^{[K]}$. With eq. (B.2), the
microcanonical partition function can be written in the form

$\displaystyle\Gamma=\int\Delta^{[K]}\,d^{DN}Q\,.$ (B.4)

One can show that

$\displaystyle\frac{\partial\Gamma}{\partial
c_{ij}}=-\int\Delta^{[K-1]}_{ij}\Delta^{\prime}_{ij}\,d^{DN}Q\,,\qquad i\leq
j\,,$ (B.5)

where $\Delta^{\prime}_{ij}$ denotes the derivative of $\Delta_{ij}$ w.r.t.
its argument. Furthermore, one can derive the following relations, where
$Q_{\mu,k}$ is the $k^{\textrm{th}}$ component of the $\mu^{\textrm{th}}$
dumbbell,

$\displaystyle\sum_{\mu=1}^{N}Q_{\mu,k}\frac{\partial\Delta_{ij}}{\partial
Q_{\mu,k}}=\left(\delta_{ik}\hat{c}_{jk}+\delta_{jk}\hat{c}_{ik}\right)\Delta^{\prime}_{ij}\,,\qquad
i\leq j\,,$ (B.6)

of which there are three non-zero cases:

$\displaystyle i=j=k:$
$\displaystyle\sum_{\mu=1}^{N}Q_{\mu,i}\frac{\partial\Delta_{ii}}{\partial
Q_{\mu,i}}=2\left(\Delta^{\prime}_{ii}c_{ii}-\Delta_{ii}\right)\,,$ (B.7)
$\displaystyle i<j,k=i:$
$\displaystyle\sum_{\mu=1}^{N}Q_{\mu,i}\frac{\partial\Delta_{ij}}{\partial
Q_{\mu,i}}=\Delta^{\prime}_{ij}c_{ij}-\Delta_{ij}\,,$ (B.8) $\displaystyle
i<j,k=j:$
$\displaystyle\sum_{\mu=1}^{N}Q_{\mu,j}\frac{\partial\Delta_{ij}}{\partial
Q_{\mu,j}}=\Delta^{\prime}_{ij}c_{ij}-\Delta_{ij}\,.$ (B.9)

In calculating eqs. (B.7)–(B.9), we have made use of the identity
$x\delta^{\prime}(x)=-\delta(x)$, with $x=\hat{c}_{ij}-c_{ij}$, which implies
$(\hat{c}_{ij}-c_{ij})\delta^{\prime}(\hat{c}_{ij}-c_{ij})=-\delta(\hat{c}_{ij}-c_{ij})$,
from which it follows
$\hat{c}_{ij}\Delta^{\prime}_{ij}=c_{ij}\Delta^{\prime}_{ij}-\Delta_{ij}$. To
proceed, we use eq. (B.5) for $i=j$ and eq. (B.7) to derive an expression for
$2c_{ii}(\partial\Gamma/\partial c_{ii})$,

$\displaystyle 2c_{ii}\frac{\partial\Gamma}{\partial c_{ii}}$ $\displaystyle=$
$\displaystyle-\int\Delta^{[K-1]}_{ii}\left[\sum_{\mu=1}^{N}Q_{\mu,i}\frac{\partial\Delta_{ii}}{\partial
Q_{\mu,i}}+2\Delta_{ii}\right]\,d^{DN}Q$ $\displaystyle=$
$\displaystyle-2\Gamma-\int\Delta^{[K-1]}_{ii}\left[\sum_{\mu=1}^{N}Q_{\mu,i}\frac{\partial\Delta_{ii}}{\partial
Q_{\mu,i}}\right]\,d^{DN}Q\,,$

where we have used $\Delta^{[K-1]}_{ii}\Delta_{ii}=\Delta^{[K]}$. The
remaining integral is then re-written by performing an integration by parts
w.r.t. $Q_{\mu,i}$; the corresponding boundary-terms can be neglected if the
values of the components of $\bm{c}$ are finite. This leads to

$\displaystyle 2c_{ii}\frac{\partial\Gamma}{\partial c_{ii}}$ $\displaystyle=$
$\displaystyle(N-2)\Gamma+\int\Delta_{ii}\sum_{\mu=1}^{N}Q_{\mu,i}\frac{\partial\Delta^{[K-1]}_{ii}}{\partial
Q_{\mu,i}}\,d^{DN}Q\,,$

where, again, we have used $\Delta^{[K-1]}_{ii}\Delta_{ii}=\Delta^{[K]}$.
Using the product rule for calculating the derivative
$\partial\Delta^{[K-1]}_{ii}/\partial Q_{{\mu,i}}$ and rearranging terms
results in

$\displaystyle 2c_{ii}\frac{\partial\Gamma}{\partial
c_{ii}}=(N-2)\Gamma+\sum_{j=1}^{i-1}G_{ji}+\sum_{j=i+1}^{D}G^{\star}_{ij}\,,$
(B.12)

where the quantities $G_{ji}$ and $G^{\star}_{ij}$are given by

$\displaystyle G_{ji}$ $\displaystyle\equiv$
$\displaystyle\int\Delta^{[K-1]}_{ji}\sum_{\mu=1}^{N}Q_{\mu,i}\frac{\partial\Delta_{ji}}{\partial
Q_{\mu,i}}\,d^{DN}Q$ (B.13) $\displaystyle=$ $\displaystyle-
c_{ji}\frac{\partial\Gamma}{\partial c_{ji}}-\Gamma\,,\qquad j<i\,,$
$\displaystyle G^{\star}_{ij}$ $\displaystyle\equiv$
$\displaystyle\int\Delta^{[K-1]}_{ij}\sum_{\mu=1}^{N}Q_{\mu,i}\frac{\partial\Delta_{ij}}{\partial
Q_{\mu,i}}\,d^{DN}Q$ (B.14) $\displaystyle=$ $\displaystyle-
c_{ij}\frac{\partial\Gamma}{\partial c_{ij}}-\Gamma\,,\qquad i<j\,,$

where we have used eq. (B.8) and eq. (B.9) (with $i$ and $j$ interchanged),
and then eq. (B.4) and eq. (B.5). Combining eq. (B.12) with eq. (B.14) and eq.
(B.13), one obtains

$\displaystyle\sum_{j=1}^{i-1}c_{ji}\frac{\partial\Gamma}{\partial
c_{ji}}+2c_{ii}\frac{\partial\Gamma}{\partial
c_{ii}}+\sum_{j=i+1}^{D}c_{ij}\frac{\partial\Gamma}{\partial
c_{ij}}=(N-D-1)\Gamma\,,$ (B.15)

which is a differential equation for $\Gamma$. To solve this equation,
consider the ansatz

$\displaystyle\Gamma=\Gamma_{0}\left(\det\bm{c}\right)^{\nu}\,.$ (B.16)

For calculating the partial derivatives of this ansatz w.r.t. $c_{ij}$, the
following needs to be kept in mind. On the one hand, since $\bm{c}$ is
symmetric, there are only $K$ independent variables, rather than $D^{2}$,
which is in line with the strategy adopted, e.g., in eq. (B.2) and eq. (B.3).
This implies that there are only $K$ partial derivatives to be calculated in
eq. (B.15), namely w.r.t. $c_{ij}$ with $1\leq i\leq j\leq D$. On the other
hand, the tensor $\bm{c}$ contains elements $c_{ij}$ and $c_{ji}=c_{ij}$.
Hence, for $i\neq j$, one finds for any function $f(\bm{c})$: $\partial
f(\bm{c})/\partial c_{ij}=[\partial f(\bm{c})/\partial c_{ij}]_{\rm no-
sym}+[\partial f(\bm{c})/\partial c_{ji}]_{\rm no-sym}$, where “no-sym”
emphasizes that the corresponding derivative is taken without enforcing the
symmetry of $\bm{c}$, i.e., considering all $D^{2}$ components of $\bm{c}$ as
independent variables. For the derivative of ansatz (B.16) one thus obtains

$\displaystyle\frac{\partial\Gamma}{\partial
c_{kl}}=(2-\delta_{kl})\nu\Gamma\left(\bm{c}^{-1}\right)_{kl}\,,\qquad k\leq
l\,,$ (B.17)

where the factor $(2-\delta_{kl})$ originates from the fact that there are
only $K$ independent components in $\bm{c}$, $c_{kl}$ with $k\leq l$;
particularly, for $k<l$, the variable $c_{kl}$ appears at two off-diagonal
positions in the full, symmetric matrix $\bm{c}$. With this, it can be shown
that the ansatz (B.16) is indeed a solution of the differential equation
(B.15) if

$\displaystyle\nu=\frac{N-D-1}{2}\,.$ (B.18)

The $\bm{c}$-independent prefactor $\Gamma_{0}$ can not be determined with
this approach. However, since $\Gamma_{0}$ results only in an additive
contribution to the Helmholtz free energy $\Psi$, it turns out to be
irrelevant for the dynamics of the conformation tensor $\bm{c}$.

###  B.2 Procedure 2: Geometry

We begin with a geometrical interpretation of the microcanonical partition-
function $\Gamma(\bm{c})$ defined via eqs. (21) and (22). Substituting from
eq. (21) back into the full partition function eq. (20) we find:

$\displaystyle\Xi$ $\displaystyle=$ $\displaystyle\int
e^{-\Phi(\bm{c})/(k_{\rm{B}}T)}\,\Gamma(\bm{c})\,d^{K}c$ (B.19)
$\displaystyle=$ $\displaystyle\int
e^{-\Phi(\hat{\bm{c}})/(k_{\rm{B}}T)}\,d^{DN}Q.$

Comparing the two lines of this equation we recognise that we may interpret
the differential quantity $\Gamma(\bm{c})\,d^{K}c$ as being the volume, within
the $DN$ dimensional $\\{\bm{Q}_{\mu}\\}$-space, that is within an increment
$d^{K}c$ of $\bm{c}$ (i.e. where the $K$ independent components of $\bm{c}$
are each varied within an interval $dc_{\alpha}$ of their base value).
Evaluation of the dependence of this volume on $\bm{c}$ yields
$\Gamma(\bm{c})$ up to a constant prefactor. Since this derivation requires
some visual imagination, we first demonstrate how this may be reasoned in the
specific case $D=3$ before indicating how the calculation may be generalised
to arbitrary $D$.

####  B.2.1 Three dimensions

For $D=3$, $\\{\bm{Q}_{\mu}\\}$-space is a $3N$ dimensional space spanned by
the vectors $\\{\bm{Q}_{\mu}\\}_{\mu=1,\ldots,N}$. For convenience, we specify
a location within this space via the $3N$ dimensional vector
$(\mathbf{X},\mathbf{Y},\mathbf{Z})$, where
$\mathbf{X}=(x_{1},x_{2},x_{3}\ldots,x_{N})$ is the $N$-dimensional vector of
the $x$-components of $\\{\bm{Q}_{\mu}\\}_{\mu=1,\ldots,N}$ (and similarly for
$\mathbf{Y}$ and $\mathbf{Z}$).

We also note that, for $D=3$, there are $K=6$ independent components of the
symmetric $\bm{c}$ tensor: $c_{xx}$, $c_{yy}$, $c_{zz}$, $c_{xy}$, $c_{xz}$
and $c_{yz}$.

We proceed by evaluating the dependence on $\bm{c}$ of two separate volumes.
We first evaluate $V_{\mathrm{const}}$, which is the $(3N-6)$-dimensional
subvolume of $\\{\bm{Q}_{\mu}\\}$-space within which $\hat{\bm{c}}$ (given in
terms of $\\{\bm{Q}_{\mu}\\}$ by eq. 5) is held exactly equal to $\bm{c}$.
Then, for each point within $V_{\mathrm{const}}$, we find the volume $dV_{6}$
which is the $6$-dimensional subvolume swept out by varying $c_{xx}$ by
$dc_{xx}$, $c_{yy}$ by $dc_{yy}$, etc. Such excursions must all be
perpendicular to the subvolume $V_{\mathrm{const}}$ (in the
$\\{\bm{Q}_{\mu}\\}$-space), because contours of fixed $c_{xx}$ are
perpendicular to the gradient direction of $c_{xx}$ (etc.). So, we can
evaluate $\Gamma(\bm{c})\,d^{6}\bm{c}$ as the product of $V_{\mathrm{const}}$
and $dV_{6}$:

$\Gamma(\bm{c})\,d^{6}\bm{c}=V_{\mathrm{const}}\ dV_{6}\,.$

Evaluation of these volumes is most straightforward in the co-ordinate frame
in which $\bm{c}$ is diagonalised (which can always be done since $\bm{c}$ is
symmetric). In the diagonal frame, $\bm{c}$ takes values $\lambda_{x}$,
$\lambda_{y}$ and $\lambda_{z}$ along the diagonal, but is zero in the off
diagonal components. In terms of the $N$-dimensional vectors $\mathbf{X}$,
$\mathbf{Y}$ and $\mathbf{Z}$ these constraints can be expressed as:

$\displaystyle\left|\mathbf{X}\right|^{2}=\Lambda_{x}\mathrm{,~{}}\left|\mathbf{Y}\right|^{2}$
$\displaystyle=\Lambda_{y}\mathrm{,~{}}\left|\mathbf{Z}\right|^{2}=\Lambda_{z},$
$\displaystyle\mathbf{X}\cdot\mathbf{Y}=0\mathrm{,~{}}\mathbf{Y}\cdot\mathbf{Z}$
$\displaystyle=0\mathrm{,~{}}\mathbf{X}\cdot\mathbf{Z}=0,$

where $\Lambda_{i}=N\lambda_{i}$, i.e. the vectors $\mathbf{X}$, $\mathbf{Y}$
and $\mathbf{Z}$ are restricted to the surfaces of $N$-dimensional
hyperspheres of radii $\sqrt{\Lambda_{x}}$, $\sqrt{\Lambda_{y}}$, and
$\sqrt{\Lambda_{z}}$ respectively, whilst simultaneously being held to be
mutually perpendicular.

The volume $V_{\mathrm{const}}$ is the $(3N-6)$-dimensional volume swept out
by rotating the $\mathbf{X}$, $\mathbf{Y}$ and $\mathbf{Z}$ vectors subject to
the above constraints, the rotations being within the $N$-dimensional space of
these vectors. A brief analogy may help at this stage: the surface of a sphere
of radius $r$ is found by summing up small tiles formed by varying the polar
co-ordinate angles $\theta$ and $\phi$ by small increments $d\theta$ and
$d\phi$. The distance along the sphere surface moved during such increments is
$rd\theta$ and $rd\phi$ (multiplied by a geometric factor $\sin\theta$ which
is irrelevant to the scaling with $r$). So, the scaling of surface area with
$r$ can be found from the product of these lengths, $r^{2}d\theta d\phi$.
Likewise, the volume $V_{\mathrm{const}}$ is the sum over a tiling of small
incremental volumes, made by rotating the vectors $\mathbf{X}$, $\mathbf{Y}$
and $\mathbf{Z}$ by small angles $d\theta_{k}$ in each available direction,
whilst keeping their length fixed and retaining their mutually perpendicular
orientation. Since $\mathbf{X}$ has $N$ dimensions, there are in total $(N-1)$
directions in which $\mathbf{X}$ could be rotated whilst keeping the length of
$\mathbf{X}$ fixed. One such rotation will rotate the $\mathbf{X}$ vector
towards the $\mathbf{Y}$ vector. In this case, the $\mathbf{Y}$ vector must
also rotate by the same angle, so as to maintain the perpendicular condition
$\mathbf{X}\cdot\mathbf{Y}=0$. Hence, for rotation angle $d\theta_{1}$,
$\mathbf{X}$ rotates so that its end sweeps out a length
$dl_{X}=\sqrt{\Lambda_{x}}d\theta_{1}$ perpendicular to $\mathbf{X}$ (in the
direction of $\mathbf{Y}$). Likewise $\mathbf{Y}$ also rotates so that its end
sweeps out a length $dl_{Y}=\sqrt{\Lambda_{y}}d\theta_{1}$ perpendicular to
$\mathbf{Y}$ (in the direction of $-\mathbf{X}$). These changes in
$\mathbf{X}$ and $\mathbf{Y}$ result in a total length moved in
$\\{\bm{Q}_{\mu}\\}$-space which is

$\displaystyle dl_{1}$ $\displaystyle=\sqrt{dl_{X}^{2}+dl_{Y}^{2}}$
$\displaystyle=\sqrt{\Lambda_{x}+\Lambda_{y}}d\theta_{1}.$

Similarly a second rotation direction of the $\mathbf{X}$ vector is available,
towards the $\mathbf{Z}$ vector (in which case $\mathbf{Z}$ must also rotate).
Likewise, a third rotation carries $\mathbf{Y}$ towards $\mathbf{Z}$ without
rotating $\mathbf{X}$. By similar arguments, the total length moved in
$\\{\bm{Q}_{\mu}\\}$-space for these is:

$\displaystyle dl_{2}$ $\displaystyle=$
$\displaystyle\sqrt{\Lambda_{x}+\Lambda_{z}}d\theta_{2},$ $\displaystyle
dl_{3}$ $\displaystyle=$
$\displaystyle\sqrt{\Lambda_{y}+\Lambda_{z}}d\theta_{3}.$

These three rotation directions having been dealt with, there remain $(N-3)$
further rotation directions available for the $\mathbf{X}$ vector, all of
which allow $\mathbf{X}$ to remain perpendicular to both $\mathbf{Y}$ and
$\mathbf{Z}$. For each of these rotations of $\mathbf{X}$, the length moved in
$\\{\bm{Q}_{\mu}\\}$-space is

$dl_{k}=\sqrt{\Lambda_{x}}d\theta_{k},\,k=4\ldots N.$

Similarly, there are $(N-3)$ rotations available to each of $\mathbf{Y}$ and
$\mathbf{Z}$, each of which leaves the other vectors unchanged, giving
lengths:

$\displaystyle dl_{k}$ $\displaystyle=$
$\displaystyle\sqrt{\Lambda_{y}}d\theta_{k},\,k=(N+1)\ldots(2N-3),$
$\displaystyle dl_{k}$ $\displaystyle=$
$\displaystyle\sqrt{\Lambda_{z}}d\theta_{k},\,k=(2N-2)\ldots(3N-6).$

A single “tile” in the volume $V_{\mathrm{const}}$ is obtained by sweeping
through each of the above $(3N-6)$ incremental rotations. The total volume
$V_{\mathrm{const}}$ is obtained by adding together all such tiles as the
vectors are rotated. We require only the dependence of $V_{\mathrm{const}}$ on
$\bm{c}$ (ignoring prefactors). Hence, by taking the product of the lengths
swept out by a set of incremental rotations, we obtain:

$\prod_{k=1}^{3N-6}dl_{k}\sim V_{\mathrm{const}}\prod_{k=1}^{3N-6}d\theta_{k}$

where

$\displaystyle V_{\mathrm{const}}$
$\displaystyle\sim\left(\sqrt{\Lambda_{x}\Lambda_{y}\Lambda_{z}}\right)^{(N-3)}\sqrt{\Lambda_{x}+\Lambda_{y}}\sqrt{\Lambda_{x}+\Lambda_{z}}\sqrt{\Lambda_{y}+\Lambda_{z}}.$
(B.20)

We now turn to the volume $dV_{6}$, the $6$-dimensional subvolume swept out by
incrementing $c_{xx}$ by $dc_{xx}$, $c_{yy}$ by $dc_{yy}$, etc. away from a
single point within $V_{\mathrm{const}}$. We first calculate the length in
$\\{\bm{Q}_{\mu}\\}$-space traversed by incrementing $c_{xx}$ by $dc_{xx}$ in
a direction perpendicular to the contour of constant $c_{xx}$. We note that

$c_{xx}=\mathbf{X}\cdot\mathbf{X}/N,$

so the direction perpendicular to the contour of constant $c_{xx}$ is

$\nabla c_{xx}=(2\mathbf{X}/N,\mathbf{0},\mathbf{0})\,.$

Moving a distance $dl_{xx}$ in this direction changes $c_{xx}$ by

$\displaystyle dc_{xx}$ $\displaystyle=\left|\nabla c_{xx}\right|dl_{xx}$
$\displaystyle=(2\left|\mathbf{X}\right|/N)dl_{xx}$
$\displaystyle=(2\sqrt{\Lambda_{x}}/N)dl_{xx}.$

Inverting this, the distance moved in $\\{\bm{Q}_{\mu}\\}$-space when
incrementing $c_{xx}$ by $dc_{xx}$ is

$dl_{xx}=\frac{N}{2\sqrt{\Lambda_{x}}}dc_{xx}$

and similarly for $dc_{yy}$ and $dc_{zz}$.

We next note that

$c_{xy}=\mathbf{X}\cdot\mathbf{Y}/N,$

so the direction perpendicular to the contour of constant $c_{xy}$ is

$\nabla c_{xy}=(\mathbf{Y}/N,\mathbf{X}/N,\mathbf{0})$

Moving a distance $dl_{xy}$ in this direction changes $c_{xy}$ by

$\displaystyle dc_{xy}$ $\displaystyle=\left|\nabla c_{xy}\right|dl_{xy}$
$\displaystyle=\left(\sqrt{\left|\mathbf{X}\right|^{2}+\left|\mathbf{Y}\right|^{2}}/N\right)dl_{xy}$
$\displaystyle=(\sqrt{\Lambda_{x}+\Lambda_{y}}/N)dl_{xy}.$

Inverting this, the distance moved in $\\{\bm{Q}_{\mu}\\}$-space when
incrementing $c_{xy}$ by $dc_{xy}$ is

$dl_{xy}=\frac{N}{\sqrt{\Lambda_{x}+\Lambda_{y}}}dc_{xy}$

and similarly for $dc_{xz}$ and $dc_{yz}$.

Hence, multiplying these six lengths together,

$dV_{6}\sim\frac{N^{6}d^{6}\bm{c}}{\sqrt{\Lambda_{x}\Lambda_{y}\Lambda_{z}}\sqrt{\Lambda_{x}+\Lambda_{y}}\sqrt{\Lambda_{x}+\Lambda_{z}}\sqrt{\Lambda_{y}+\Lambda_{z}}}.$
(B.21)

Evaluating $\Gamma(\bm{c})\,d^{6}\bm{c}=V_{\mathrm{const}}\ dV_{6}$, and hence
$\Gamma(\bm{c})$, we find that all factors of form
$\sqrt{\Lambda_{x}+\Lambda_{y}}$ cancel, and we are left with

$\displaystyle\Gamma(\bm{c})$ $\displaystyle\sim$ $\displaystyle
N^{6}\left(\Lambda_{x}\Lambda_{y}\Lambda_{z}\right)^{(N-4)/2}$ (B.22)
$\displaystyle\sim$ $\displaystyle
N^{3N/2}\left(\lambda_{x}\lambda_{y}\lambda_{z}\right)^{(N-4)/2}.$

But $\mathrm{det}\bm{c}=\lambda_{x}\lambda_{y}\lambda_{z}$, and so:

$\Gamma(\bm{c})\sim N^{3N/2}\left(\mathrm{det}\bm{c}\right)^{(N-4)/2}$ (B.23)

with dependence on $\mathrm{det}\bm{c}$ as required.

####  B.2.2 Generalisation to $D$ dimensions

Most of the above formalism carries directly over to $D$ dimensions. In
calculating $V_{\mathrm{const}}$, each vector of type $\mathbf{X}$ has a total
number of $(N-1)$ rotations available (whilst preserving length), but $(D-1)$
of these rotate the vector towards one of the others. So, the total number of
rotations available for each vector, which do not also require rotating one of
the other vectors, is $(N-D)$. The number of rotations requiring two vectors
to rotate (i.e. of the form $\mathbf{X}$ to $\mathbf{Y}$, $\mathbf{Y}$ to
$\mathbf{Z}$ etc., but avoiding double counting) is $M=D(D-1)/2$. So,
$V_{\mathrm{const}}$ is of form:

$\displaystyle V_{\mathrm{const}}\sim$
$\displaystyle\left(\Lambda_{x}\Lambda_{y}\Lambda_{z}\ldots\right)^{(N-D)/2}$
$\displaystyle\times\sqrt{\Lambda_{x}+\Lambda_{y}}\sqrt{\Lambda_{x}+\Lambda_{z}}\sqrt{\Lambda_{y}+\Lambda_{z}}\ldots,$

where there are $M$ terms of form $\sqrt{\Lambda_{x}+\Lambda_{y}}$.

In calculating $dV_{K}$, there are $D$ diagonal constraints and $M=D(D-1)/2$
off-diagonal constraints in $\bm{c}$ (a total of $K=D(D+1)/2$). Each diagonal
constraint gives a factor of form $\sqrt{\Lambda_{x}}$ in the denominator of
$dV_{K}$, whilst each off diagonal constraint gives a factor of form
$\sqrt{\Lambda_{x}+\Lambda_{y}}$. Hence:

$dV_{K}\sim\frac{N^{K}d^{K}\bm{c}}{\sqrt{\Lambda_{x}\Lambda_{y}\Lambda_{z}\ldots}\sqrt{\Lambda_{x}+\Lambda_{y}}\sqrt{\Lambda_{x}+\Lambda_{z}}\sqrt{\Lambda_{y}+\Lambda_{z}}\ldots}.$

In the product $\Gamma(\bm{c})\,d^{K}\bm{c}=V_{\mathrm{const}}\ dV_{K}$, all
$M$ factors of form $\sqrt{\Lambda_{x}+\Lambda_{y}}$ are present once in the
numerator, and once in the denominator, and so cancel. Hence:

$\displaystyle\Gamma(\bm{c})\,d^{K}\bm{c}$
$\displaystyle=V_{\mathrm{const}}.dV_{K}$ $\displaystyle\sim
N^{K}\left(\Lambda_{x}\Lambda_{y}\Lambda_{z}\ldots\right)^{(N-D-1)/2}d^{K}\bm{c}$
$\displaystyle\sim
N^{DN/2}\left(\lambda_{x}\lambda_{y}\lambda_{z}\ldots\right)^{(N-D-1)/2}d^{K}\bm{c}$
$\displaystyle\sim
N^{DN/2}\left(\mathrm{det}\bm{c}\right)^{(N-D-1)/2}d^{K}\bm{c}.$ (B.24)

###  B.3 Procedure 3: Scaling

In this procedure, scaling arguments are employed for the calculation of the
microcanonical partition function $\Gamma$, eq. (22), in real space and in
reciprocal space, respectively.

####  B.3.1 Scaling in real space

Similar to the procedure described in sec. B.2, it is again chosen to describe
the $\\{\bm{Q}_{\mu}\\}$-space in terms of the $N$-dimensional vectors
$\mathbf{X}_{i}$, $1\leq i\leq D$, where the $\mu$-th component of
$\mathbf{X}_{i}$ equals the $i$-th component of $\bm{Q}_{\mu}$.

In what follows, we consider the coordinate system in which the conformation
tensor $\bm{c}$ is diagonal, with eigenvalues $\lambda_{i}$ $(1\leq i\leq D)$.
The microcanonical partition function $\Gamma$, eq. (22) with instantaneous
conformation tensor $\hat{\bm{c}}=\hat{\bm{c}}(\bm{Q})$, can be written in the
$\\{\mathbf{X}_{i}\\}$-representation as

$\displaystyle\Gamma(\bm{c})=\int\,\delta^{(K)}\left(\hat{\bm{c}}(\mathbf{X})-\bm{c}\right)\,\prod_{i=1}^{D}\prod_{\mu=1}^{N}dX_{i,\mu}\,,$
(B.25)

with $\hat{\bm{c}}=\hat{\bm{c}}(\mathbf{X})$ given by
$\hat{c}_{ij}=\mathbf{X}_{i}\cdot\mathbf{X}_{j}/N$. In view of the conditions
$\hat{c}_{ii}=c_{ii}=\lambda_{i}$, the integral (B.25) is re-written by
introducing $\mathbf{X}_{i}=\sqrt{\lambda_{i}}\tilde{\mathbf{X}}_{i}$. Making
use of the property $\delta(ax)=(1/a)\delta(x)$ for $a>0$, one obtains from
eq. (B.25),

$\displaystyle\Gamma(\bm{c})=\Gamma_{1}(\bm{c})\,\Gamma_{2}(\bm{c})\,\Gamma_{3}\,,$
(B.26)

with

$\displaystyle\Gamma_{1}(\bm{c})$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{D}\prod_{\mu=1}^{N}\sqrt{\lambda_{i}}=(\det\bm{c})^{N/2}\,,$
(B.27) $\displaystyle\Gamma_{2}(\bm{c})$ $\displaystyle=$
$\displaystyle\prod_{1\leq i\leq j\leq
D}\frac{1}{\sqrt{\lambda_{i}\lambda_{j}}}$ (B.28) $\displaystyle=$
$\displaystyle\sqrt{\left(\prod_{1\leq i,j\leq
D}\frac{1}{\sqrt{\lambda_{i}\lambda_{j}}}\right)\left(\prod_{1\leq i\leq
D}\frac{1}{\lambda_{i}}\right)}$ $\displaystyle=$
$\displaystyle\sqrt{(\det\bm{c})^{-D}(\det\bm{c})^{-1}}$ $\displaystyle=$
$\displaystyle(\det\bm{c})^{(-D-1)/2}\,,$ $\displaystyle\Gamma_{3}$
$\displaystyle=$
$\displaystyle\int\,\delta^{(K)}\left(\hat{\bm{c}}(\tilde{\mathbf{X}})-\bm{1}\right)\,\prod_{i=1}^{D}\prod_{\mu=1}^{N}d\tilde{X}_{i,\mu}\,,$
(B.29)

where $\Gamma_{1}$ comes from the substitution of variables in the volume
element, and $\Gamma_{2}$ comes from the substitution of variables in
$\delta^{(K)}$. It is to be noted that the integral $\Gamma_{3}$ does not
depend on $\bm{c}$. Rather, we find $\Gamma_{3}=\Gamma(\bm{1})$. Therefore,
one obtains for the $\bm{c}$-dependence of the microcanonical partition
function

$\displaystyle\Gamma(\bm{c})=(\det\bm{c})^{(N-D-1)/2}\Gamma(\bm{1})\,.$ (B.30)

####  B.3.2 Scaling in reciprocal space

An alternative derivation for $\Gamma(\bm{c})$ follows by explicitly enforcing
the constraints using Fourier transforms, and then performing a rescaling. We
begin with the microcanonical partition-function $\Gamma$ defined in eq. (22),

$\displaystyle\Gamma(\bm{c})$ $\displaystyle=$ $\displaystyle\int
d^{DN}Q\,\delta^{(K)}\left(\bm{c}-\hat{\bm{c}}\right)\,,$ (B.31)

where $\hat{\bm{c}}$ is the symmetric number-averaged conformation tensor
given by eq. (5). The $\delta$-function ensures that only configurations
$\\{\bm{Q}_{\mu}\\}$ that satisfy $\bm{c}=\hat{\bm{c}}$ are included in
$\Gamma(\bm{c})$. To evaluate $\Gamma(\bm{c})$ we choose a coordinate system
for the $\bm{Q}_{\mu}$ comprising the eigenvectors
$\\{\bm{\widehat{e}}_{a}\\}$ and eigenvalues $\lambda_{a}$ of $\bm{c}$. The
constraints in the Dirac $\delta$-function then reduce to

$\displaystyle\lambda_{a}$ $\displaystyle=$
$\displaystyle\bm{\widehat{e}}_{a}\cdot\hat{\bm{c}}\cdot\bm{\widehat{e}}_{a}\quad(a=1,\dots
D)\quad D\,\textrm{constraints},$ (B.32) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\bm{\widehat{e}}_{b}\cdot\hat{\bm{c}}\cdot\bm{\widehat{e}}_{c}\quad(b<c)\quad
L\,\textrm{constraints},$ (B.33)

where there are $L=D(D-1)/2$ zero off-diagonal elements. Hence we introduce
$D$ Dirac $\delta$-functions for the diagonal constraints, and $L$
$\delta$-functions for the off-diagonal constraints. We enforce these with
Fourier transforms, leading to

$\displaystyle\Gamma(\bm{c})$ $\displaystyle=$
$\displaystyle\int\frac{d^{D}\\!u}{(2\pi)^{D}}\int\frac{d^{L}\\!v}{(2\pi)^{L}}\int
d^{DN}Q$
$\displaystyle\exp\left[i\sum_{a=1}^{D}u_{a}\left(\lambda_{a}-\bm{\widehat{e}}_{a}\cdot\hat{\bm{c}}\cdot\bm{\widehat{e}}_{a}\right)-i\sum^{D}_{\begin{subarray}{c}b,c=1\\\
b<c\end{subarray}}v_{\alpha}\bm{\widehat{e}}_{b}\cdot\hat{\bm{c}}\cdot\bm{\widehat{e}}_{c}\right],$

where the second sum is over all $L$ distinct pairs of eigenvectors ($b<c$).
Inserting the definition for $\hat{\bm{c}}$ (eq. (5)) and interchanging
$\sum_{\mu}$ and $\sum_{a}$ in the exponential, $\Gamma(\bm{c})$ can be
written as

$\Gamma(\bm{c})=\frac{2^{L}}{(2\pi)^{K}}\int
d^{D}\\!u\,e^{i\bm{u}\cdot\bm{\lambda}}\int d^{L}\\!v\int d^{DN}Q\\\
\exp\left[-\frac{i}{N}\sum_{\mu=1}^{N}\bm{Q}_{\mu}\cdot\left\\{\mathsf{U}+\mathsf{V}\right\\}\cdot\bm{Q}_{\mu}\right],$
(B.35)

where the matrices $\mathsf{U}$ and $\mathsf{V}$ are constructed from the
constraint fields $u_{a}$ and $v_{\alpha}$ as follows (represented in the
basis $\\{\bm{\widehat{e}}_{a}\\}$) for $D=3$

$\displaystyle\mathsf{U}$ $\displaystyle=\left(\begin{matrix}u_{1}&0&0\\\
0&u_{2}&0\\\ 0&0&u_{3}\end{matrix}\right),$ $\displaystyle\mathsf{V}$
$\displaystyle=\left(\begin{matrix}0&v_{1}&v_{2}\\\ v_{1}&0&v_{3}\\\
v_{2}&v_{3}&0\end{matrix}\right),$ (B.36)

and we have rescaled $\bm{v}$ by a factor of 2. Next, we scale the $\bm{u}$
and $\bm{v}$ integrals according to

$\displaystyle u_{a}$ $\displaystyle=\frac{\bar{u}_{a}}{\lambda_{a}}$
$\displaystyle d^{D}\\!u$
$\displaystyle=\frac{d^{D}\bar{u}}{\left(\det\bm{c}\right)}$ $\displaystyle
a=1\ldots D$ (B.37) $\displaystyle v_{\alpha}$
$\displaystyle=\frac{\bar{v}_{\alpha}}{\sqrt{\lambda_{a}\lambda_{b}}},$
$\displaystyle d^{L}\\!v$
$\displaystyle=\frac{d^{L}\bar{v}}{\left(\det\bm{c}\right)^{(D-1)/2}}$
$\displaystyle\stackrel{{\scriptstyle\alpha=1\ldots L}}{{\scriptstyle
a<b\in(1,D)}}$ (B.38)

where we recognize $\det\bm{c}=\prod_{a=1}^{D}\lambda_{a}$. The scale factors
for the $v_{\alpha}$ correspond to the $u_{a}$ in its row and column. To see
the scaling in the measure for $\bar{v}$, note that each $v$ picks up a factor
in the denominator proportional to $\lambda$. Hence the denominator scales as
$\lambda^{L}\sim\lambda^{D(D-1)/2}\sim(\det\bm{c})^{(D-1)/2}$, leading to the
denominator above in eq. (B.38). Hence,

$\displaystyle\Gamma(\bm{c})$
$\displaystyle=\frac{2^{L}}{(2\pi)^{K}\left(\det\bm{c}\right)^{(D+1)/2}}\int
d^{D}\\!\bar{u}\,e^{i{\bar{\bm{u}}\cdot\bm{1}}}\int d^{L}\\!\bar{v}\int
d^{DN}\,Q$
$\displaystyle\qquad\times\exp\left[-\frac{i}{N}\sum_{\mu=1}^{N}\bm{Q}_{\mu}\cdot\left\\{{\mathsf{U}}+{\mathsf{V}}\right\\}\cdot\bm{Q}_{\mu},\right],$
(B.39)

where $\bm{1}=(1,1,\ldots)$ and now, for $(D=3)$, $\mathsf{U}+\mathsf{V}$ is
expressed in the form

$\displaystyle{\mathsf{U}}+{\mathsf{V}}$
$\displaystyle=\left(\begin{matrix}\displaystyle\frac{\bar{u}_{1}}{\sqrt{\lambda_{1}^{2}}}&\displaystyle\frac{\bar{v}_{1}}{\sqrt{\lambda_{1}\lambda_{2}}}&\displaystyle\frac{\bar{v}_{2}}{\sqrt{\lambda_{1}\lambda_{3}}}\\\
\displaystyle\frac{\bar{v}_{1}}{\sqrt{\lambda_{1}\lambda_{2}}}&\displaystyle\frac{\bar{u}_{2}}{\sqrt{\lambda_{2}^{2}}}&\displaystyle\frac{\bar{v}_{3}}{\sqrt{\lambda_{2}\lambda_{3}}}\\\
\displaystyle\frac{\bar{v}_{2}}{\sqrt{\lambda_{1}\lambda_{3}}}&\displaystyle\frac{\bar{v}_{3}}{\sqrt{\lambda_{2}\lambda_{3}}}&\displaystyle\frac{\bar{u}_{3}}{\sqrt{\lambda_{3}^{2}}}\end{matrix}\right).$
(B.40)

We finally rescale $\bm{Q}_{\mu}=\bar{\bm{Q}}_{\mu}\cdot\bm{\Lambda}$, where
$\bm{\Lambda}$ is the diagonal matrix
$\Lambda_{aa}=\sqrt{\lambda}_{a},\Lambda_{ab}=0(a\neq b)$. Hence, we find

$\displaystyle\Gamma(\bm{c})$
$\displaystyle=\frac{2^{L}}{(2\pi)^{K}}\,\left(\det\bm{c}\right)^{(N-D-1)/2}\int
d^{D}\\!\bar{u}\,e^{i\bar{\bm{u}}\cdot\bm{1}}\int d^{L}\\!\bar{v}\int
d^{DN}\\!\bar{Q}$ (B.41)
$\displaystyle\qquad\times\exp\left[-\frac{i}{N}\sum_{\mu=1}^{N}\bar{\bm{Q}}_{\mu}\cdot\left\\{\bar{\mathsf{U}}+\bar{\mathsf{V}}\right\\}\cdot\bar{\bm{Q}}_{\mu}\right],$
$\displaystyle\equiv\left(\det\bm{c}\right)^{(N-D-1)/2}\,\,{\cal I},$ (B.42)

where the integral ${\cal I}$ is independent of the conformation tensor
$\bm{c}$, and $\bar{\mathsf{U}}+\bar{\mathsf{V}}$ is equal to (B.40) without
the factors $1/\sqrt{\lambda_{a}\lambda_{b}}$.

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