# Extension of Kirkwood-Buff theory: Partial enthalpies, fluctuations of
energy density, temperature, and pressure, and solute-induced effects in a
mixture solvent

Akira Onuki111e-mail<EMAIL_ADDRESS>(corresponding author)
Department of Physics, Kyoto University, Kyoto 606-8502, Japan

###### Abstract

We present a statistical mechanical theory of multi-component fluids, where we
consider the correlation functions of the number densities and the energy
density in the grand canonical ensemble. In terms of their space integrals we
express the partial volumes ${\bar{v}}_{i}$, the partial enthalpies
${\bar{H}}_{i}$, and other thermodynamic derivatives. These ${\bar{v}}_{i}$
and ${\bar{H}}_{i}$ assume simple forms for binary mixtures and for ternary
mixtures with a dilute solute. They are then related to the space-dependent
thermal fluctuations of the temperature and the pressure. The space averages
of these fluctuations are those introduced by Landau and Lifshits in the
isothermal-isobaric ($T$-$p$) ensemble. We also give expressions for the long-
range (nonlocal) correlations in the canonical and $T$-$p$ ensembles, which
are inversely proportional to the system volume. For a mixture solvent, we
examine the solvent-induced solute-solute attraction and the osmotic enthalpy
changes due to the solute doping using the correlation function integrals.
Keywords: Fluctuation solution theory, Partial enthalpies, Energy density
fluctuation, Temperature-pressure fluctuations, Solute-solute interactions

1\. Introduction
The Kirkwood-Buff (KB) integrals $G_{ij}$ Buff ; Naim2 ; Naim3 ; Naim ; Oc
have been used extensively in the investigation of the molecular interactions
in multi-component fluids. For species $i$ and $j$, $G_{ij}$ are the space
integrals of $g_{ij}(r)-1$, where $g_{ij}(r)$ are the radial distribution
functions. They have been determined experimentally by the small-angle
scattering of $x$-ray and neutrons from binary mixtures Nishi ; Misawa ; YKoga
; Lep , where the combination $G_{11}+G_{22}-2G_{12}$ represents asymmetry of
the molecular interactions and its large size indicates enhancement of the
concentration fluctuation Bagchi1 ; Ruck ; Koga . Ghosh and Bagchi Bagchi1
numerically studied such clustering dynamically. In biology Pa ; Tim ; Smith ;
Smith1 ; Record ; Shimizu ; Mochi , attention has been paid to the action of a
cosolvent on the solubility and stability of proteins in a primary solvent. In
such solutions, the solute interacts with the two solvent species differently
and the difference $G_{32}-G_{31}$ represents the strength of preferential
solvation, where the index 3 refers to the solute. This difference grows with
increasing the solute size above the solvent ones.

The partial volumes ${\bar{v}}_{i}$ have been measured (including those of
ions in water) Jones ; Naka ; Craig . They are characteristic volumes per
particle satisfying the sum rule ${\sum}_{i}{\bar{v}}_{i}n_{i}=1$, where
$n_{i}$ are the average number densities. They can be expressed in terms of
$G_{ij}$ Buff ; Naim2 ; Naim ; Naim3 ; Oc . On the other hand, the partial
enthalpies ${\bar{H}}_{i}$ are theoretically difficult quantities beyond the
Kirkwood-Buff theory. They are defined as the enthalpy change upon addition of
a single particle of the $i$-th species at fixed temperature $T$ and pressure
$p$ in a macroscopic solution. Thus, ${\bar{H}}_{i}$ were measured by
calorimetric methods Hepler ; Nil ; YKoga1 ; Maham . In chemical reactions,
the reaction enthalpy is a linear combination of the partial enthalpies of the
reactant and product particles Landau-s ; Callen . As numerical calculations
of ${\bar{H}}_{i}$, Frenkel’s group Frenkel  used a particle-insertion
technique, while Vlugt’s group Bedeaux ; Vlugt ; Vlugt1 ; VlugtF  used some
thermodynamic relations of the energy density. In this paper, we consider the
correlation functions
$\langle{\delta{\hat{e}}({\mbox{\boldmath$r$}})\delta{\hat{n}}_{i}({\mbox{\boldmath$0$}})}\rangle$
of the fluctuation of the energy density
$\delta{\hat{e}}({\mbox{\boldmath$r$}})$ and those of the number densities
$\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})$ in the grand-canonical ensemble.
We write their space integrals as $n_{i}J_{i}$ and express the partial
enthalpies in terms of $J_{i}$ and $G_{ij}$.

In this paper, we introduce space-dependent temperature and pressure
fluctuations Landau-s ; Onukibook , written as
$\delta{\hat{T}}({\mbox{\boldmath$r$}})$ and
$\delta{\hat{p}}({\mbox{\boldmath$r$}})$, respectively. They are linear
combinations of $\delta{\hat{e}}({\mbox{\boldmath$r$}})$ and
$\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})$, where the coefficients in front
of $\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})$ are thermodynamic derivatives
closely related to ${\bar{H}}_{i}$. We also introduce the fluctuation of the
volume fraction
$\delta{\hat{\phi}}({\mbox{\boldmath$r$}})={\sum}_{i}{\bar{v}}_{i}\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})$
using ${\bar{v}}_{i}$ Kirk3 ; Oka1 , which can be expressed as a linear
combination of $\delta{\hat{T}}({\mbox{\boldmath$r$}})$ and
$\delta{\hat{p}}({\mbox{\boldmath$r$}})$. We then calculate the correlation
function integrals among $\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})$,
$\delta{\hat{e}}({\mbox{\boldmath$r$}})$,
$\delta{\hat{T}}({\mbox{\boldmath$r$}})$,
$\delta{\hat{p}}({\mbox{\boldmath$r$}})$,
$\delta{\hat{\phi}}({\mbox{\boldmath$r$}})$, and the microscopically defined
local stress tensor ${\hat{\Pi}}_{\alpha\beta}({\mbox{\boldmath$r$}})$
($\alpha,\beta=x,y,z$). Landau and Lifshitz Landau-s  introduced temperature
and pressure fluctuations, $\Delta T$ and $\Delta p$, in a finite system in
the isothermal-isobaric ($T$-$p$) ensemble Kubo ; Binderbook ; Fre , which are
the space averages of $\delta{\hat{T}}({\mbox{\boldmath$r$}})$ and
$\delta{\hat{p}}({\mbox{\boldmath$r$}})$ in the container. We also give
expressions for the long-range (nonlocal) correlations in the $T$-$p$
ensemble, which are well known for the canonical ensemble Lebo .

For binary systems, ${\bar{v}}_{i}$ are known to have simple expressions in
terms of $G_{jk}$ Buff ; Naim ; Naim2 ; Oc . In this paper, we express
${\bar{H}}_{i}$ in simple forms in terms of $J_{j}$ and $G_{jk}$. For ternary
solutions composed of a mixture solvent and a dilute solute, we present
expressions of ${\bar{v}}_{3}$ and ${\bar{H}}_{3}$ for the solute in terms of
$J_{i}$ and $G_{ij}$ up to the linear order in the solute density $n_{3}$.
These expressions are further simplified in the infinite-dilution limit
$n_{3}\to 0$. The critical behaviors of ${\bar{v}}_{i}$ and ${\bar{H}}_{i}$
are known unambiguously from their correlation function expressions.

For ternary solutions, we present a theory on the solvent-mediated, solute-
solute attraction by calculating $G_{33}$ for the solute. It arises from the
solute-induced solvent density changes, which are divided into those of
$\delta{\hat{\phi}}({\mbox{\boldmath$r$}})$ Chan ; Ben ; Lev ; Oo ; Jiang ;
Oka1 ; Bagchi  and those of the concentration deviation
$\delta{\hat{X}}({\mbox{\boldmath$r$}})=[n_{1}\delta{\hat{n}}_{2}({\mbox{\boldmath$r$}})-n_{2}\delta{\hat{n}}_{1}({\mbox{\boldmath$r$}})]/n^{2}$
Hori ; Zemb1 ; Oka1 . As a related problem, Asakura and Oosawa presented an
attractive potential between colloidal particles due to polymer depletion in
solvent-polymer-colloid mixtures Oosawa ; Vrij ; Binder ; Lek1 . Finally, we
examine the thermodynamics of a ternary solution in osmotic equilibrium with a
reservoir at fixed solvent chemical potentials $\mu_{1}$ and $\mu_{2}$. In
this situation, the osmotic pressure $\Pi$ has been investigated Mc ; Koga ;
Ruck . We further calculate the enthalpy change $\Delta H$ as a function of
the solute density $n_{3}$.

The organization of this paper is as follows. In Section 2, we will present
thermodynamic relations of the partial quantities. In Section 3, we will
introduce the correlation function integrals. In Section 4, we will discuss
the fluctuations in the temperature and the pressure. In Section 5, the
partial quantities in binary mixtures will be calculated. In Section 6, the
partial quantities and the solvent-mediated interaction will be discussed in
ternary mixtures. In section 7, the thermodynamic of ternary mixtures in
osmotic equilibrium will be investigated.

2\. Thermodynamics of multi-component fluids
2.1. Partial quantities at fixed $p$ and $T$
We consider an equilibrium fluid mixture composed of $m$ components without
electrostatic interactions in a macroscopic volume $V$. We assume no applied
field such as gravity and electric field and neglect the boundary effects. In
this section, we present fundamental thermodynamic relations needed in
Sections 3-7.

Let us consider an extensive quantity $Q$ as a function of the temperature
$T$, the pressure $p$, and the particle numbers $N_{i}=Vn_{i}$ ($1\leq i\leq
m)$ with $n_{i}$ being the average densities. Its partial quantities per
particle are defined by

${\bar{Q}}_{i}={\bigg{(}}\frac{\partial{Q}}{\partial{N_{i}}}{\bigg{)}}_{T,p,N_{j\neq
i}},$ (1)

where we fix $T$, $p$, and $N_{j}$ with $j\neq i$ in the derivative. The
extensivity of $Q$ means the relation $Q(T,p,\\{\lambda N\\})=\lambda
Q(T,p,\\{N\\})$ for $N_{i}\lambda\gg 1$, which yields the sum rule,

$Q={\sum}_{i}{\bar{Q}}_{i}N_{i}=V{\sum}_{i}{\bar{Q}}_{i}n_{i}.$ (2)

For the volume $V$ and the total entropy $S$, the partial volumes
${\bar{v}}_{i}$ and the partial entropies ${\bar{S}}_{i}$ are written as

${\bar{v}}_{i}={\bigg{(}}\frac{\partial{V}}{\partial{N_{i}}}{\bigg{)}}_{T,p,N_{j\neq
i}},~{}~{}{\bar{S}}_{i}={\bigg{(}}\frac{\partial{S}}{\partial{N_{i}}}{\bigg{)}}_{T,p,N_{j\neq
i}}.$ (3)

For the Gibbs free energy $G$, its partial quantities are equal to the
chemical potentials $\mu_{i}$ from its differential equation,

$dG={\sum}_{i}\mu_{i}dN_{i}-SdT+Vdp.$ (4)

Therefore, from the enthalpy relation $H=G+TS$, the partial enthalpies are
given by

${\bar{H}}_{i}=\mu_{i}+T{\bar{S}}_{i}.$ (5)

Varying $(\\{N\\},T,p)$ we have the differential equation,

$dQ={\sum}_{i}{\bar{Q}}_{i}dN_{i}+{\bigg{(}}\frac{\partial{Q}}{\partial{T}}{\bigg{)}}_{p,\\{N\\}}dT+{\bigg{(}}\frac{\partial{Q}}{\partial{p}}{\bigg{)}}_{T,\\{N\\}}dp.$
(6)

For $Q=V$ and $S$ the above relation is written as

$\displaystyle dV={\sum}_{i}{\bar{v}}_{i}dN_{i}+V\alpha_{p}dT-V\kappa_{T}dp,$
(7) $\displaystyle dS={\sum}_{i}{\bar{S}}_{i}dN_{i}+VC_{p}dT/T-V\alpha_{p}dp,$
(8)

where $\alpha_{p}$ is the isobaric thermal expansivity, $\kappa_{T}$ is the
isothermal compressibility, and $C_{p}$ is the isobaric heat capacity per unit
volume. In the thermodynamic limit $V\to\infty$, we have

$\displaystyle\alpha_{p}=-\frac{1}{n}{\bigg{(}}\frac{\partial{n}}{\partial{T}}{\bigg{)}}_{p,\\{x\\}},~{}~{}~{}\kappa_{T}=\frac{1}{n}{\bigg{(}}\frac{\partial{n}}{\partial{p}}{\bigg{)}}_{T,\\{x\\}},~{}~{}~{}{C_{p}}={nT}{\bigg{(}}\frac{\partial{\sigma}}{\partial{T}}{\bigg{)}}_{p,\\{x\\}},$
(9)

where $n=\sum_{i}n_{i}=N/V$ and $N={\sum}_{i}N_{i}$. Here, $\sigma=S/N$ is the
entropy per particle and is related to the entropy density by $s=S/V=n\sigma$,
so $VC_{p}=T(\partial S/\partial T)_{p,\\{N\\}}=NT(\partial\sigma/\partial
T)_{p,\\{x\\}}$. Note that the thermodynamic derivatives of intensive
quantities at fixed $\\{N\\}$ are those at fixed molar concentrations
$\\{x\\}$. In this paper, $\\{x\\}$ denotes the set of
$x_{i}=N_{i}/N=n_{i}/n$, while $\\{n\\}$ denotes the set of $n_{i}$.

We further consider the density variable $q\equiv Q/V$. From Eqs.(6) and (7),
it obeys

$dq={\sum}_{i}{\bar{Q}}_{i}dn_{i}+{\bigg{(}}\frac{\partial{q}}{\partial{T}}{\bigg{)}}_{p,\\{x\\}}dT+{\bigg{(}}\frac{\partial{q}}{\partial{p}}{\bigg{)}}_{T,\\{x\\}}dp+q(\alpha_{p}dT-\kappa_{T}dp).$
(10)

Since $q=1$ for $Q=V$, Eq.(10) yields the differential equation Oka1 ,

${\sum}_{i}{\bar{v}}_{i}dn_{i}+\alpha_{p}dT-\kappa_{T}dp=0,$ (11)

where the first term is the incremental change in the particle volume fraction
(see Eqs.(38) and (47)). Then, we have two expressions for ${\bar{v}}_{i}$:

${\bar{v}}_{i}=\kappa_{T}{\bigg{(}}\frac{\partial{p}}{\partial{n_{i}}}{\bigg{)}}_{T,n_{j\neq
i}},~{}~{}{\bar{v}}_{i}=-\alpha_{p}{\bigg{(}}\frac{\partial{T}}{\partial{n_{i}}}{\bigg{)}}_{p,n_{j\neq
i}}.$ (12)

The first relation is convenient for dilute mixtures, since $p$ can be
expanded with respect to the solute densities (see Section 6). The second
relation follows from the first since $\alpha_{p}=\kappa_{T}(\partial
p/\partial T)_{\\{n\\}}$ (see Eq.(27)).

Next, we define a symmetric matrix $\\{I_{ij}\\}$ and its inverse matrix
$\\{I^{ij}\\}$ as

$I_{ij}={k_{B}T}{\bigg{(}}\frac{\partial{n_{j}}}{\partial{\mu_{i}}}{\bigg{)}}_{T,\mu_{k\neq
i}},~{}I^{ij}=\frac{1}{k_{B}T}{\bigg{(}}\frac{\partial{\mu_{j}}}{\partial{n_{i}}}{\bigg{)}}_{T,n_{k\neq
i}},$ (13)

where $T$ is fixed and ${\sum}_{k}I_{ik}I^{kj}=\delta_{ij}$. Using the Gibbs-
Duhem relation in the first relation in Eq.(12), we can relate
$\\{{\bar{v}}\\}$ and $\\{{n}\\}$ as

${\bar{v}}_{i}=k_{B}T\kappa_{T}{\sum}_{j}I^{ij}n_{j},~{}~{}{\sum}_{j}I_{ij}{\bar{v}}_{j}=k_{B}T\kappa_{T}n_{i}.$
(14)

The isothermal compressibility $\kappa_{T}$ is then expressed as

$k_{B}T{\kappa_{T}}=1{\big{/}}\big{[}{\sum}_{i,j}n_{i}n_{j}I^{ij}\big{]}={\sum}_{i,j}{\bar{v}}_{i}{\bar{v}}_{j}I_{ij}.$
(15)

We note that Eq.(4) yields the well-known expression,

${\bar{v}}_{i}={\bigg{(}}\frac{\partial{\mu_{i}}}{\partial{p}}{\bigg{)}}_{T,\\{x\\}}=k_{B}T{\sum}_{j}I^{ij}{\bigg{(}}\frac{\partial{n_{j}}}{\partial{p}}{\bigg{)}}_{T,\\{x\\}}.$
(16)

which is equivalent to Eq.(12) from $(\partial n_{j}/\partial
p)_{T,\\{x\\}}=n_{j}\kappa_{T}$ for any $j$. Then, Eq.(4) also gives

${\bar{S}}_{i}=-{\bigg{(}}\frac{\partial{\mu_{i}}}{\partial{T}}{\bigg{)}}_{p,\\{x\\}},~{}~{}{\bar{H}}_{i}=-T^{2}{\bigg{(}}\frac{\partial{(\mu_{i}/T)}}{\partial{T}}{\bigg{)}}_{p,\\{x\\}}.$
(17)

Here, we have $(\partial w/\partial T)_{p,\\{x\\}}=(\partial w/\partial
T)_{\\{n\\}}-\alpha_{p}n(\partial w/\partial n)_{T,\\{x\\}}$ for any $w$.

2.2. Derivatives of energy density with respect to $n_{i}$
We next introduce the thermodynamic derivatives of the internal energy density
$e$ with respect to the number densities $n_{i}$:

$\displaystyle
M_{i}={\bigg{(}}\frac{\partial{e}}{\partial{n_{i}}}{\bigg{)}}_{T,n_{j\neq
i}}=-T^{2}{\bigg{(}}\frac{\partial{(\mu_{i}/T)}}{\partial{T}}{\bigg{)}}_{\\{n\\}},$
(18) $\displaystyle
R_{i}={\bigg{(}}\frac{\partial{e}}{\partial{n_{i}}}{\bigg{)}}_{p,n_{j\neq
i}}=M_{i}-\frac{1}{\alpha_{p}}C_{V}{\bar{v}}_{i},$ (19)

where we fix $T$ in $M_{i}$ and $p$ in $R_{i}$. In Eq.(18) use is made of
$(\partial s/\partial n_{i})_{T,n_{j\neq i}}=-(\partial\mu_{i}/\partial
T)_{\\{n\\}}$, while in Eq.(19) $R_{i}-M_{i}$ follows from the second relation
in Eq.(12). The $C_{V}$ is the isochoric heat capacity per unit volume. Then,
$e$ obeys the differential equations,

$\displaystyle de={\sum}_{i}M_{i}d{n}_{i}+C_{V}{dT}$ (20)
$\displaystyle={\sum}_{i}R_{i}d{n}_{i}+T\alpha_{s}{dp}.$ (21)

where $T\alpha_{s}=(\partial e/\partial p)_{\\{n\\}}=C_{V}(\partial T/\partial
p)_{\\{n\\}}$ (see Eqs.(26) and (27)). These equations follow from the
fundamental equation $de=Tds+{\sum}_{i}\mu_{i}dn_{i}$. From Eqs.(17)-(19) the
differences ${\bar{H}}_{i}-{M}_{i}$ and ${\bar{H}}_{i}-{R}_{i}$ are
proportional to ${\bar{v}}_{i}$ as

${\bar{H}}_{i}=M_{i}+T\beta_{V}{\bar{v}}_{i}=R_{i}+T\beta_{s}{\bar{v}}_{i}.$
(22)

See the definitions of $C_{V}$, $\alpha_{s}$, $\beta_{V}$, and $\beta_{s}$ in
Eq.(26). We can also derive $\beta_{s}-\beta_{V}=C_{V}/T\alpha_{p}$ from
Eq.(27), so Eq.(22) is consistent with Eq.(19).

From Eq.(22) the enthalpy density $h=H/V={\sum}_{i}{\bar{H}}_{i}n_{i}$ is
expressed as

$h=e+p={\sum}_{i}M_{i}n_{i}+T\beta_{V}={\sum}_{i}R_{i}n_{i}+T\beta_{s},$ (23)

in terms of $M_{i}$ and $R_{i}$. Thus, ${\bar{H}}_{i}$ can also be expressed
as

$\displaystyle{\bar{H}}_{i}=M_{i}+{\bar{v}}_{i}\big{(}h-{\sum}_{j}M_{j}n_{j}\big{)}$
(24) $\displaystyle=R_{i}+{\bar{v}}_{i}\big{(}h-{\sum}_{j}R_{j}n_{j}\big{)}.$
(25)

Vlugt Vlugt  derived Eq.(24) and used it to calculate ${\bar{H}}_{i}$ for
binary mixtures.

2.3. Thermodynamic derivatives at fixed $\\{x\\}$
In addition to $\alpha_{p}$, $\kappa_{T}$, and $C_{p}$ in Eq.(9), we define
the following derivatives at fixed $\\{x\\}$:

$\displaystyle
C_{V}=T{\bigg{(}}\frac{\partial{s}}{\partial{T}}{\bigg{)}}_{\\{n\\}}=\frac{1}{\gamma}C_{p},~{}~{}~{}\beta_{V}={\bigg{(}}\frac{\partial{p}}{\partial{T}}{\bigg{)}}_{\\{n\\}},~{}~{}\beta_{s}={\bigg{(}}\frac{\partial{p}}{\partial{T}}{\bigg{)}}_{\sigma,\\{x\\}},$
$\displaystyle\alpha_{s}=\frac{1}{n}{\bigg{(}}\frac{\partial{n}}{\partial{T}}{\bigg{)}}_{\sigma,\\{x\\}},~{}~{}\kappa_{s}=\frac{1}{n}{\bigg{(}}\frac{\partial{n}}{\partial{p}}{\bigg{)}}_{\sigma,\\{x\\}}=\frac{1}{\gamma}\kappa_{T}.$
(26)

Here, $\alpha_{s}$ is the adiabatic thermal expansivity and $\kappa_{s}$ is
the adiabatic compressibility. The $\alpha_{s}$, $\beta_{s}$, and $\kappa_{s}$
are the isentropic derivatives at fixed $\sigma=s/n$ and $\\{x\\}$, while
$\gamma=C_{p}/C_{V}=\kappa_{T}/\kappa_{s}$ is the specific heat ratio. We list
useful thermodynamic identities,

$\displaystyle\beta_{s}=\alpha_{s}/\kappa_{s}=C_{p}/T\alpha_{p},~{}~{}\beta_{V}=\alpha_{p}/\kappa_{T}=C_{V}/T\alpha_{s},$
$\displaystyle{\gamma-1}=\gamma{\beta_{V}}/\beta_{s}={\alpha_{p}}/{\alpha_{s}}=T\alpha_{p}^{2}/\kappa_{T}C_{V},$
(27)

which are well known for pure fluids and still hold for multi-component fluids
at fixed $\\{x\\}$.

3\. Correlation functions
3.1. Grand-canonical variances
For any space-dependent fluctuating variables $\hat{\cal
A}({\mbox{\boldmath$r$}})$ and $\hat{\cal B}({\mbox{\boldmath$r$}})$, we
consider the space integral of their equal-time correlation function in the
bulk region Onukibook ,

$\langle{\hat{\cal A}:\hat{\cal B}}\rangle=\int
d{\mbox{\boldmath$r$}}\langle{\delta{\hat{\cal
A}}({\mbox{\boldmath$r$}})\delta{{\hat{\cal
B}}}({\mbox{\boldmath$0$}})}\rangle$ (28)

where $\delta{\hat{\cal A}}={\hat{\cal A}}-\langle{{\hat{\cal A}}}\rangle$,
$\delta{\hat{\cal B}}={\hat{\cal B}}-\langle{{\hat{\cal B}}}\rangle$,
$\langle{\cdots}\rangle$ denotes the equilibrium average, and the positions
$r$ and $0$ are located far from the boundaries. The correlation
$\langle{\delta{\hat{\cal A}}({\mbox{\boldmath$r$}})\delta{{\hat{\cal
B}}}({\mbox{\boldmath$0$}})}\rangle$ is taken in the grand-canonical
($T$-$\mu$) ensemble at fixed $V$ Kubo , so it decays rapidly if
$r=|{\mbox{\boldmath$r$}}|$ much exceeds the molecular correlation lengths. We
call $\langle{\hat{\cal A}:\hat{\cal B}}\rangle$ the $T$-$\mu$ variance of
$\hat{\cal A}$ and $\hat{\cal B}$. In this paper, space-dependent quantities
with a caret are thermally fluctuating variables.

The local number densities and the local energy density have well-defined
microscopic expressions in terms of the particle positions and momenta, so
they are written as

${\hat{n}}_{i}({\mbox{\boldmath$r$}})=n_{i}+\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}}),~{}~{}{\hat{e}}({\mbox{\boldmath$r$}})=e+\delta{\hat{e}}({\mbox{\boldmath$r$}}),$
(29)

where $n_{i}$ and $e$ are the equilibrium averages and the second terms are
the fluctuating deviations. Then, as will be shown in Appendix A, the
thermodynamic derivatives $I_{ij}$ in Eq.(13) are equal to the density
variances:

$I_{ij}=\langle{{\hat{n}}_{i}:{\hat{n}}_{j}}\rangle=n_{i}\delta_{ij}+n_{i}n_{j}G_{ij}.$
(30)

where $G_{ij}$ are the Kirkwood-Buff (KB) integrals Buff ; Oc ; Naim2 ; Naim ;
Naim3 . On the other hand, the space integrals of the direct correlation
functions $c_{ij}(r)$ are equal to $\delta_{ij}/n_{i}-I^{ij}$.

In this paper, we consider the integrals of the grand-canonical correlation
functions of the energy density and the number densities:

$n_{i}J_{i}=\langle{{\hat{e}}:{\hat{n}}_{i}}\rangle=\int
d{\mbox{\boldmath$r$}}\langle{\delta{\hat{e}}({\mbox{\boldmath$r$}})\delta{\hat{n}}_{i}({\mbox{\boldmath$0$}})}\rangle,$
(31)

where $J_{i}$ remains finite as $n_{i}\to 0$. Then, $n_{i}J_{i}$ and
$\langle{{\hat{e}}:{\hat{e}}}\rangle$ are expressed as

$\displaystyle n_{i}J_{i}={\sum}_{j}M_{j}I_{ji}$ (32)
$\displaystyle={\sum}_{j}R_{j}I_{ji}+k_{B}T^{2}\alpha_{s}n_{i}$ (33)
$\displaystyle={\sum}_{j}{\bar{H}}_{j}I_{ji}-k_{B}T^{2}\alpha_{p}n_{i},$ (34)
$\displaystyle\langle{{\hat{e}}:{\hat{e}}}\rangle=k_{B}T^{2}C_{V}+{\sum}_{i,j}M_{i}M_{j}I_{ij},$
(35)

in terms of $M_{i}$, $R_{i}$, and ${\bar{H}}_{i}$. Here, Eqs.(32) and (35)
will be derived in Appendix A, while Eqs.(33) and (34) follow from Eq.(32)
with the aid of Eqs.(22) and (27). Note that Eq.(32) is rewritten as
$M_{i}={\sum}_{j}n_{j}J_{j}I^{ji}$, which was derived by Karavias and Myers
Myers  in their theory on heats of adsorption. In Eq.(35), the second term is
rewritten as
${\sum}_{i,j}\langle{{\hat{e}}:{\hat{n}}_{i}}\rangle\langle{{\hat{e}}:{\hat{n}}_{j}}\rangle
I^{ij}$, so $C_{V}$ is expressed in terms of the $T$-$\mu$ variances. Mishin
Mishin1  obtained an expression equivalent to Eq.(35) for the $T$-$\mu$
ensemble.

From Eq.(34) ${\bar{H}}_{i}$ are written in terms of $J_{i}$ and $I_{ij}$ as

$\displaystyle{\bar{H}}_{i}=T\beta_{V}{\bar{v}}_{i}+{\sum}_{j}n_{j}J_{j}I^{ji}$
$\displaystyle=h{\bar{v}}_{i}+{\sum}_{j}n_{j}J_{j}\big{[}I^{ji}-{{\bar{v}}_{j}{\bar{v}}_{i}}/{k_{B}T\kappa_{T}}\big{]}.$
(36)

where the second line follows from Eqs.(14) and (23). The first line of
Eq.(36) further yields

$T\alpha_{p}=T\beta_{V}\kappa_{T}=h\kappa_{T}-{\sum}_{i}n_{i}{\bar{v}}_{i}J_{i}/k_{B}T,$
(37)

where $\alpha_{p}$ is written in terms of the $T$-$\mu$ variances as well as
$\kappa_{T}$ in Eq.(15). See simpler expressions of $\alpha_{p}$,
$\kappa_{T}$, and $C_{p}$ in Eqs.(67), (A.15), and (A16) for binary mixtures.

3.2. Fluctuations of volume fraction and concentrations
We introduce the fluctuations of the volume fraction and the concentrations as

$\displaystyle\delta{\hat{\phi}}({\mbox{\boldmath$r$}})={\sum}_{i}{\bar{v}}_{i}{\hat{n}}_{i}({\mbox{\boldmath$r$}})-1={\sum}_{i}{\bar{v}}_{i}\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}}),$
(38)
$\displaystyle\delta{\hat{X}}_{i}({\mbox{\boldmath$r$}})=\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})/n-{n_{i}}\delta{\hat{n}}({\mbox{\boldmath$r$}})/n^{2},$
(39)

where $n={\sum}_{i}n_{i}$ and ${\sum}_{i}\delta{\hat{X}}_{i}=0$. See another
expression of $\delta{\hat{\phi}}$ in Eq.(47). From Eqs.(14), (15), and (37)
we find simple variance relations,

$\displaystyle\langle{{\hat{\phi}}:{\hat{\phi}}}\rangle=k_{B}T\kappa_{T},~{}~{}\langle{{\hat{\phi}}:{\hat{n}}_{i}}\rangle=k_{B}T\kappa_{T}n_{i},$
$\displaystyle\langle{{\hat{\phi}}:{\hat{X}}_{i}}\rangle=0,~{}~{}\langle{{\hat{\phi}}:{\hat{e}}}\rangle=k_{B}T\kappa_{T}(h-T\beta_{V}).$
(40)

The $\delta{\hat{\phi}}$ and $\delta{\hat{X}}_{i}$ were used originally by
Kirkwood and Goldberg Kirk3  and recently by our group Oka1  in calculating
the scattering intensities (see Eq.(72)).

The fluctuation $\delta{\hat{\phi}}({\mbox{\boldmath$r$}})$ is small in nearly
incompressible fluids with $\epsilon_{\rm in}\equiv k_{B}Tn\kappa_{T}\ll 1$,
where $\epsilon_{\rm in}$ represents the degree of compressibility. For
example, $\epsilon_{\rm in}\sim 0.05$ for water-alcohol mixtures in the
ambient condition (at $T\cong 300$ K and $p\cong 1$ atm) Kubota .

3.3. Correlations in the canonical ensemble
In finite systems, the correlation functions can have a long-range (nonlocal)
part in the canonical ensemble Lebo ; Naim1 ; Kremer ; Eg ; Ev ; Onuki-h ,
where the particle numbers $N_{i}$ and the volume $V$ are fixed. That is, for
any space-dependent fluctuating variables ${\hat{\cal
A}}({\mbox{\boldmath$r$}})$ and ${\hat{\cal B}}({\mbox{\boldmath$r$}})$, their
space correlation in the canonical ensemble $\langle{\cdots}\rangle_{\rm ca}$,
and that in the $T$-$\mu$ one $\langle{\cdots}\rangle$ are related by Lebo

$\langle{\delta{\hat{\cal A}}({\mbox{\boldmath$r$}})\delta{{\hat{\cal
B}}}({\mbox{\boldmath$0$}})}\rangle_{\rm ca}=\langle{\delta{\hat{\cal
A}}({\mbox{\boldmath$r$}})\delta{{\hat{\cal
B}}}({\mbox{\boldmath$0$}})}\rangle-{\sum}_{i,j}D^{ij}_{{\cal A}{\cal
B}}I_{ij}/V,$ (41)

where $r$ and $0$ are located far from the boundaries. In the right hand side,
the first term is the short-ranged $T$-$\mu$ correlation, while the second
term is a nonlocal constant proportional to $V^{-1}$ and is small for large
$V$. The coefficients $D^{ij}_{{\cal A}{\cal B}}$ are defined by

$D^{ij}_{{\cal A}{\cal B}}={\bigg{(}}\frac{\partial{\langle{\hat{\cal
A}}\rangle}}{\partial{n_{i}}}{\bigg{)}}_{T,{n_{k\neq
i}}}{\bigg{(}}\frac{\partial{\langle{\hat{\cal
B}}\rangle}}{\partial{n_{j}}}{\bigg{)}}_{T,{n_{\ell\neq
j}}}={\sum}_{k,\ell}\langle{{\hat{\cal
A}}:{{\hat{n}}_{k}}}\rangle\langle{{\hat{\cal B}}:{{\hat{n}}_{\ell}}}\rangle
I^{ki}I^{\ell j},$ (42)

where we use Eq.(A5). Then, integrating Eq.(41) in the cell, we find

$\frac{1}{V}\int_{V}d{\mbox{\boldmath$r$}}_{1}\int_{V}d{\mbox{\boldmath$r$}}_{2}\langle{\delta{\hat{\cal
A}}({\mbox{\boldmath$r$}}_{1})\delta{{\hat{\cal
B}}}({\mbox{\boldmath$r$}}_{2})}\rangle_{\rm ca}=\langle{{\hat{\cal
A}}:{{\hat{\cal B}}}}\rangle-{\sum}_{i,j}\langle{{\hat{\cal
A}}:{{\hat{n}}_{i}}}\rangle\langle{{\hat{\cal B}}:{{\hat{n}}_{j}}}\rangle
I^{ij},$ (43)

where the second term is independent of $V$ and can be comparable to the
first. Note that the finite-size correction due to the boundaries is of order
$L^{-1}$ in Eq.(43), where $L=V^{1/3}$ is the linear dimension of the cell. A
simple derivation of Eq.(41) was given in Appendix C in Ref.Onuki-h . More
results on the canonical correlations will be given around Eq.(52). The
nonlocal correlations in the $T$-$p$ ensemble will be studied in Section 4.5.

For $\delta{\hat{n}}_{i}$ and $\delta{\hat{e}}$ we rewrite Eq.(41) as

$\displaystyle\langle{\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})\delta{\hat{n}}_{j}({\mbox{\boldmath$0$}})}\rangle_{\rm
ca}=\langle{\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})\delta{\hat{n}}_{j}({\mbox{\boldmath$0$}})}\rangle-
I_{ij}/V,$
$\displaystyle\langle{\delta{\hat{e}}({\mbox{\boldmath$r$}})\delta{\hat{n}}_{i}({\mbox{\boldmath$0$}})}\rangle_{\rm
ca}=\langle{\delta{\hat{e}}({\mbox{\boldmath$r$}})\delta{\hat{n}}_{i}({\mbox{\boldmath$0$}})}\rangle-
n_{i}J_{i}/V,$
$\displaystyle\langle{\delta{\hat{e}}({\mbox{\boldmath$r$}})\delta{\hat{e}}({\mbox{\boldmath$0$}})}\rangle_{\rm
ca}=\langle{\delta{\hat{e}}({\mbox{\boldmath$r$}})\delta{\hat{e}}({\mbox{\boldmath$0$}})}\rangle-{\sum}_{i,j}M_{i}M_{j}I_{ij}/V,$
(44)

where we use Eqs.(18) and (32). Here, the space integrals of the first two
relations vanish and that of the third one is $k_{B}T^{2}C_{V}$ from Eq.(35),
which should hold in the canonical ensemble.

4\. Fluctuations of hydrodynamic variables
4.1. Fluctuations of temperature and pressure
Using the thermodynamic equations (20) and (21), we define the local
fluctuations of the temperature and the pressure as linear combinations of
$\delta{\hat{e}}({\mbox{\boldmath$r$}})$ and
$\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})$ Onukibook :

$\displaystyle\delta{\hat{T}}({\mbox{\boldmath$r$}})=\big{[}\delta{\hat{e}}({\mbox{\boldmath$r$}})-{\sum}_{i}M_{i}\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})\big{]}/C_{V},$
(45)
$\displaystyle\delta{\hat{p}}({\mbox{\boldmath$r$}})=\big{[}\delta{\hat{e}}({\mbox{\boldmath$r$}})-{\sum}_{i}R_{i}\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})\big{]}/T\alpha_{s},$
(46)

where $M_{i}$ and $R_{i}$ are defined in Eqs.(18) and (19). These fluctuations
are superimposed on the thermodynamic temperature and pressure $T$ and $p$.
They consist of the Fourier components with wavelengths longer than the
molecular correlation lengths, so they obey the Gaussian distributions in the
linear regime. These definitions can be used for any statistical ensembles. We
remove $\delta{\hat{e}}$ from Eqs.(45) and (46) using Eqs.(19) and (27) to
find

$\delta{\hat{\phi}}({\mbox{\boldmath$r$}})={\sum}_{i}{\bar{v}}_{i}\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})=-\alpha_{p}\delta{\hat{T}}({\mbox{\boldmath$r$}})+\kappa_{T}\delta{\hat{p}}({\mbox{\boldmath$r$}}).$
(47)

where $\delta{\hat{\phi}}({\mbox{\boldmath$r$}})$ is given in Eq.(38). Notice
that the above fluctuation equation assumes the same form as the thermodynamic
differential equation (11).

From Eqs.(32)-(35) and (40) we find the $T$-$\mu$ variances of
$\delta{\hat{T}}$ and $\delta{\hat{p}}$:

$\displaystyle\langle{{{\hat{n}}_{i}}:{\hat{T}}}\rangle=0,~{}~{}\langle{{\hat{n}}_{i}:{\hat{p}}}\rangle=k_{B}Tn_{i},~{}~{}\langle{{\hat{e}}:{\hat{T}}}\rangle=k_{B}T^{2},~{}~{}\langle{{\hat{e}}:{\hat{p}}}\rangle=k_{B}Th,$
$\displaystyle\langle{{\hat{T}}:{\hat{T}}}\rangle=k_{B}T^{2}/C_{V},~{}~{}\langle{{{\hat{p}}}:{\hat{T}}}\rangle=k_{B}T/\alpha_{s},~{}~{}\langle{{\hat{p}}:{\hat{p}}}\rangle=k_{B}T/\kappa_{s},~{}~{}$
$\displaystyle\langle{{{\hat{T}}}:{\hat{\phi}}}\rangle=\langle{{\hat{T}}:{\hat{X}}_{i}}\rangle=0,~{}~{}\langle{{{\hat{p}}}:{\hat{\phi}}}\rangle=k_{B}T,~{}~{}\langle{{\hat{p}}:{\hat{X}}_{i}}\rangle=0,$
(48)

The first relation shows that $\delta{\hat{T}}$ is orthogonal to
$\delta{\hat{n}}_{i}$, so Eqs.(45) and (48) yield Eq.(35). In Eq.(A12) in
Appendix A, we will also introduce the entropy fluctuation.

In dynamics, the Fourier components of the fluctuating hydrodynamic variables
have been assumed to obey the linearized hydrodynamic equations with random
source terms (as Langevin equations) Martin ; Mountain1 ; Bernebook , where
the wavelengths are longer than the molecular correlation lengths. Their time
correlation functions have been calculated with the Laplace transformation
method in the time range $t>0$, where their initial values at $t=0$ are given
by the variances in Eq.(48). These results are used to interpret the dynamic
light scattering.

In the vicinity of the fluid criticality, we can define $\delta{\hat{T}}$ and
$\delta{\hat{p}}$ by Eqs.(45) and (46) if the upper cutoff wave number
$\Lambda$ of the fluctuations is smaller than the inverse correlation length
$\xi^{-1}$ (after the renormalization procedure), where $\xi$ grows and the
most singular parts of $\delta{\hat{e}}$ and $\delta{\hat{n}}_{i}$ are
cancelled to vanish. We can further define them for
$\xi^{-1}\lesssim\Lambda\ll a^{-1}$ with $a$ being the typical molecular
radius if we assume a mapping relationship between fluids and Ising
ferromagnets Onukibook ; OnukiL ; Ani . Indeed, in near-critical ferromagnets,
the thermal fluctuations of the temperature and the magnetic field can be
equated to the functional derivatives of the (fluctuating) Ginzburg-Landau-
Wilson free energy (with respect to the energy density and the magnetization)
Hal ; Onukibook , where they depend on the magnetization fluctuation
nonlinearly for $\xi^{-1}\lesssim\Lambda\ll a^{-1}$ (in the course of the
renormalization).

Our definitions of $\delta{\hat{T}}({\mbox{\boldmath$r$}})$ and
$\delta{\hat{p}}({\mbox{\boldmath$r$}})$ in Eqs.(45) and (46) are natural and
rather obvious. It is worth noting that Kittel Kittel  strictly wrote
”Temperature fluctuation is an oxymoron because the consistent and consensual
definition of temperature admits no fluctuation.”

4.2. Entropy deviation in second order
Greene and Callen Callen ; Callen1  obtained the correlation moments of
general fluctuating extensive variables by calculating the entropy change
induced by them. In the same spirit, Boltzmann’s principle of equal
probability indicates that the equilibrium distribution of
$\delta{\hat{e}}({\mbox{\boldmath$r$}})$ and
$\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})$ at long wavelengths is given by

$P_{\rm eq}(\delta{\hat{e}},\delta{\hat{n}})={\rm
const.}\exp\big{[}\int_{V}d{\mbox{\boldmath$r$}}(\delta{\hat{s}})_{2}({\mbox{\boldmath$r$}})/k_{B}\big{]},$
(49)

where $(\delta{\hat{s}})_{2}({\mbox{\boldmath$r$}})$ is the second-order
deviation at position $r$ in the fluctuating entropy density in the Gaussian
approximation at long wavelengths:

$\displaystyle(\delta{\hat{s}})_{2}=s({\hat{e}},\\{{\hat{n}}\\})-s({e},\\{{n}\\})-\delta{\hat{e}}/T+{\sum}_{i}\mu_{i}\delta{\hat{n}}_{i}/T$
$\displaystyle=-{C_{V}}(\delta{\hat{T}}/T)^{2}/{2}-k_{B}{\sum}_{i,j}{I^{ij}}\delta{\hat{n}}_{i}\delta{\hat{n}}_{j}/2.$
(50)

Here, we perform the expansion of $s({\hat{e}},\\{{\hat{n}}\\})$ with respect
to $\delta{\hat{e}}={\hat{e}}-{e}$ and
$\delta{\hat{n}}_{i}={\hat{n}}_{i}-{n}_{i}$, where $s({e},\\{{n}\\})$ is the
thermodynamic entropy density and $\delta{\hat{T}}$ is defined by Eq.(45)
Landau-s . Thus, the above entropic distribution $P_{\rm eq}$ surely gives the
$T$-$\mu$ variances in Eq.(48).

4.3. Fluctuations of stress tensor and pressure
The space-dependent stress tensor
${\hat{\Pi}}_{\alpha\beta}({\mbox{\boldmath$r$}})$ ($\alpha,\beta=x,y,z$) has
a microscopic expression in terms of the particle positions and momenta Kirk1
; Onukibook . The divergence
$-{\sum}_{\beta}\partial{\hat{\Pi}}_{\alpha\beta}/\partial x_{\beta}$ is the
force density on the fluid and is equal to the time derivative of the momentum
density (without applied force field). The thermodynamic pressure $p$ is
obtained from the equilibrium average
$\langle{{\hat{\Pi}}_{\alpha\beta}}\rangle=\delta_{\alpha\beta}p$ in the bulk
region far from the boundaries.

In Appendix A, we will derive variance relations,

$\langle{{\hat{\Pi}}_{\alpha\beta}:{\hat{n}}_{i}}\rangle=\delta_{\alpha\beta}k_{B}Tn_{i},~{}~{}\langle{{\hat{\Pi}}_{\alpha\beta}:{\hat{e}}}\rangle=\delta_{\alpha\beta}k_{B}Th,$
(51)

where the right hand sides of these two relations coincide with
$\delta_{\alpha\beta}\langle{{\hat{p}}:{\hat{n}}_{i}}\rangle$ and
$\delta_{\alpha\beta}\langle{{\hat{p}}:{\hat{e}}}\rangle$, respectively, from
Eq.(48). Thus, the pressure fluctuation $\delta{\hat{p}}$ in Eq.(46) is the
projection of the diagonal part of
${\hat{\Pi}}_{\alpha\beta}-\delta_{\alpha\beta}p$ onto $\delta{\hat{e}}$ and
$\delta{\hat{n}}_{i}$ Zwan ; Mori ; Onukibook . Indeed, if we use Mori’s
linear projection operator ${\cal P}$ onto the hydrodynamic fluctuations Mori
, we have ${\cal
P}{\hat{\Pi}}_{\alpha\beta}=\delta_{\alpha\beta}(p+\delta{\hat{p}}).$ The
difference
${\hat{\Pi}}_{\alpha\beta}^{\perp}={\hat{\Pi}}_{\alpha\beta}-\delta_{\alpha\beta}(p+\delta{\hat{p}})$
is orthogonal to the hydrodynamic variables, which was examined by some
authors Wallace ; Sasa . In the linear response theory, the viscosities are
given by the space-time integral of the dynamical correlation
$\langle{{\hat{\Pi}}_{\alpha\beta}^{\perp}({\mbox{\boldmath$r$}},t){\hat{\Pi}}_{\gamma\delta}^{\perp}({\mbox{\boldmath$0$}},0)}\rangle$
Onukibook ; Mori ; Zwan .

If ${\cal{\hat{B}}}={\hat{p}}$ in Eq.(41), Eqs.(42), (46), and (47) give the
following canonical correlations:

$\displaystyle\langle{\delta{\hat{\cal
A}}({\mbox{\boldmath$r$}})\delta{\hat{p}}({\mbox{\boldmath$0$}})}\rangle_{\rm
ca}=\langle{\delta{\hat{\cal
A}}({\mbox{\boldmath$r$}})\delta{\hat{p}}({\mbox{\boldmath$0$}})}\rangle-\langle{{\hat{\cal
A}}:{\hat{\phi}}}\rangle/(V\kappa_{T}),$
$\displaystyle\int_{V}d{\mbox{\boldmath$r$}}\langle{\delta{\hat{\cal
A}}({\mbox{\boldmath$r$}})\delta{\hat{p}}({\mbox{\boldmath$0$}})}\rangle_{\rm
ca}=\beta_{V}\langle{{\hat{\cal A}}:{\hat{T}}}\rangle.$ (52)

If ${\cal{\hat{A}}}={\cal{\hat{B}}}={\hat{p}}$, we have
$\int_{V}d{\mbox{\boldmath$r$}}\langle{\delta{\hat{p}}({\mbox{\boldmath$r$}})\delta{\hat{p}}({\mbox{\boldmath$0$}})}\rangle_{\rm
ca}=k_{B}T(\gamma-1)/\kappa_{T}$ from
$D^{ij}_{{p}{p}}={\bar{v}}_{i}{\bar{v}}_{j}/\kappa_{T}^{2}$. However,
$\langle{\delta{\hat{\cal
A}}({\mbox{\boldmath$r$}})\delta{\hat{T}}({\mbox{\boldmath$0$}})}\rangle_{\rm
ca}$ has no nonlocal part for any $\hat{\cal A}({\mbox{\boldmath$r$}})$ from
$D^{ij}_{{\cal A}{T}}=0$.

4.4. Landau-Lifshitz theory
Landau and Lifshitz Landau-s  introduced fluctuations of the temperature and
the pressure, written as $\Delta T$ and $\Delta p$, which are linear
combinations of extensive changes in the volume and the energy, $\Delta V$ and
$\Delta E$. They used the $T$-$p$ distribution Kubo ; Binderbook ; Fre , where
the particle numbers $N_{i}$ are fixed but the volume fluctuation $\Delta V$
is superimposed on the average volume $V$ with the $T$-$p$ variances Landau-s
given by

$\displaystyle\langle{(\Delta
V)^{2}}\rangle=Vk_{B}T\kappa_{T},~{}~{}\langle{\Delta V\Delta
p}\rangle=-k_{B}T,$ $\displaystyle\langle{\Delta V\Delta
T}\rangle=0,~{}~{}\langle{\Delta V\Delta
E}\rangle=Vk_{B}T\kappa_{T}(T\beta_{V}-p).$ (53)

These relations hold for multi-component fluids, where use is made of the
thermodynamic derivatives at fixed $\\{x\\}$ in Eqs.(9) and (26). Here, we
note that $\Delta T$ and $\Delta p$ in the Landau-Lifshitz book are the space
averages of $\delta{\hat{T}}({\mbox{\boldmath$r$}})$ and
$\delta{\hat{p}}({\mbox{\boldmath$r$}})$ in Eqs.(45) and (46):

$\displaystyle\Delta
T=\frac{1}{V}\int_{V}d{\mbox{\boldmath$r$}}\delta{\hat{T}}({\mbox{\boldmath$r$}}),~{}~{}\Delta
p=\frac{1}{V}\int_{V}d{\mbox{\boldmath$r$}}\delta{\hat{p}}({\mbox{\boldmath$r$}}).$
(54)

We can define $\Delta T$ and $\Delta p$ by Eq.(54) for any ensembles.

With nonvanishing $\Delta V$, the particle-number changes $\Delta N_{i}$ and
the internal-energy change $\Delta E$ are expressed in the linear order as

$\Delta N_{i}=\int_{V}d{\mbox{\boldmath$r$}}\delta{\hat{n}}_{i}+n_{i}\Delta
V,~{}~{}\Delta E=\int_{V}d{\mbox{\boldmath$r$}}\delta{\hat{e}}+e\Delta V,$
(55)

With the aid of Eqs.(23), (45), and (46) we can then express $\Delta T$ and
$\Delta p$ as

$\displaystyle\Delta T=\Big{[}\Delta E+(p-T\beta_{V})\Delta
V-{\sum}_{i}M_{i}\Delta N_{i}\Big{]}/VC_{V},$ $\displaystyle\Delta
p=\Big{[}\Delta E+(p-T\beta_{s})\Delta V-{\sum}_{i}R_{i}\Delta
N_{i}\Big{]}/(VT\alpha_{s}).$ (56)

If $\Delta N_{i}=0$, these expressions are equivalent to those in the Landau-
Lifshitz book.

In the $T$-$\mu$, canonical, and $T$-$p$ ensembles, $\Delta T$ commonly
satisfies the relations,

$\langle{(\Delta T)^{2}}\rangle={k_{B}T^{2}}/{VC_{V}},~{}~{}\langle{\Delta
T\Delta E}\rangle={k_{B}T^{2}},$ (57)

which follow from Eqs.(48) and (55). The fluctuation of $\Delta T$ is small
for large $V$. We mention an experiment Lipa  and theories Long ; Schm ;
Mishin  on $\Delta T$ in a small system in contact with a heat bath.

4.5. Correlations in the $T$-$p$ ensemble
Simulations in the $T$-$p$ ensemble have been performed extensively Fre ;
Binderbook , which are powerful for investigating phase coexistence at fixed
$T$ and $p$. Since the finite-size effects are essential in any ensembles
Bedeaux ; Kremer , we here present the counterpart of Eq.(41) for the $T$-$p$
ensemble. Namely, for any space-dependent fluctuating variables ${\hat{\cal
A}}({\mbox{\boldmath$r$}})$ and ${\hat{\cal B}}({\mbox{\boldmath$r$}})$, we
relate their $T$-$p$ correlation $\langle{\cdots}\rangle_{Tp}$, $T$-$\mu$ one
$\langle{\cdots}\rangle$, and canonical one $\langle{\cdots}\rangle_{\rm ca}$
as

$\displaystyle\langle{\delta{\hat{\cal
A}}({\mbox{\boldmath$r$}})\delta{{\hat{\cal
B}}}({\mbox{\boldmath$0$}})}\rangle_{Tp}=\langle{\delta{\hat{\cal
A}}({\mbox{\boldmath$r$}})\delta{{\hat{\cal
B}}}({\mbox{\boldmath$0$}})}\rangle-{\sum}_{i,j}D^{ij}_{{\cal A}{\cal
B}}\big{[}I_{ij}-k_{B}T\kappa_{T}n_{i}n_{j}\big{]}/V$ $\displaystyle\hskip
17.07164pt=\langle{\delta{\hat{\cal
A}}({\mbox{\boldmath$r$}})\delta{{\hat{\cal
B}}}({\mbox{\boldmath$0$}})}\rangle_{\rm ca}+\langle{{\hat{\cal
A}}:{{\hat{\phi}}}}\rangle\langle{{\hat{\cal
B}}:{{\hat{\phi}}}}\rangle/(k_{B}T\kappa_{T}V),$ (58)

where we define $D^{ij}_{{\cal A}{\cal B}}$ in Eq.(42) and
$\delta{\hat{\phi}}$ in Eq.(38) and use Eq.(14). Here,
$I_{ij}=\langle{{\hat{n}}_{i}:{\hat{n}}_{j}}\rangle$ in Eq.(41) are replaced
by
$\langle{{\hat{n}}_{i}^{\prime}:{\hat{n}}_{j}^{\prime}}\rangle=I_{ij}-k_{B}T\kappa_{T}n_{i}n_{j}$
with
$\delta{\hat{n}}_{k}^{\prime}=\delta{\hat{n}}_{k}-n_{k}\delta{\hat{\phi}}$,
where $n_{k}\delta{\hat{\phi}}$ are the dilational density changes and
$\delta{\hat{n}}_{k}^{\prime}$ vanish at fixed concentrations. From
$D^{ij}_{{\cal A}{\cal T}}=0$, the correlation of
$\delta{\hat{T}}({\mbox{\boldmath$r$}})$ and any $\delta{\hat{\cal
A}}({\mbox{\boldmath$r$}})$ has no nonlocal part in the three ensembles.

With the aid of Eqs.(23), (32), and (35), Eq.(58) yields

$\displaystyle\int_{V}d{\mbox{\boldmath$r$}}\langle{\delta{\hat{n}_{i}}({\mbox{\boldmath$r$}})\delta{{\hat{n}_{j}}}({\mbox{\boldmath$0$}})}\rangle_{Tp}/k_{B}T=\kappa_{T}n_{i}n_{j},$
$\displaystyle\int_{V}d{\mbox{\boldmath$r$}}\langle{\delta{\hat{e}}({\mbox{\boldmath$r$}})\delta{{\hat{n}_{i}}}({\mbox{\boldmath$0$}})}\rangle_{Tp}/k_{B}T=(h-T\beta_{V})\kappa_{T}n_{i},$
$\displaystyle\int_{V}d{\mbox{\boldmath$r$}}\langle{\delta{\hat{e}}({\mbox{\boldmath$r$}})\delta{{\hat{e}}}({\mbox{\boldmath$0$}})}\rangle_{Tp}/k_{B}T=TC_{V}+(h-T\beta_{V})^{2}\kappa_{T}.$
(59)

We can then derive Eq.(53) using Eq.(55). Here, with replacement
$h-T\beta_{V}\to p-T\beta_{V}$, the right hand side of the third relation
becomes equal to $\langle{(\Delta E)^{2}}\rangle_{Tp}/V$ Landau-s ; Kubo ;
Mishin1 . Also $\langle{(\Delta p)^{2}}\rangle V$ is given by
$k_{B}T/\kappa_{s}$ in the $T$-$\mu$ and $T$-$p$ ensembles and by
$k_{B}T(1/\kappa_{s}-1/\kappa_{T})$ in the canonical one from Eqs.(48), (58),
and (52), respectively.

For one-component fluids, the second term in Eq.(58) vanishes, so there is no
nonlocal correlation in the $T$-$p$ ensemble, which is in accord with a
previous simulation Velasco . For binary mixtures, the nonlocal part in
Eq.(58) is rewritten as

$\langle{\delta{\hat{\cal A}}({\mbox{\boldmath$r$}})\delta{{\hat{\cal
B}}}({\mbox{\boldmath$0$}})}\rangle_{Tp}-\langle{\delta{\hat{\cal
A}}({\mbox{\boldmath$r$}})\delta{{\hat{\cal
B}}}({\mbox{\boldmath$0$}})}\rangle=-{\bigg{(}}\frac{\partial{\langle{\hat{\cal
A}}\rangle}}{\partial{X}}{\bigg{)}}_{T,p}{\bigg{(}}\frac{\partial{\langle{{\hat{\cal
B}}}\rangle}}{\partial{X}}{\bigg{)}}_{T,p}\frac{\chi}{V}=-\frac{\langle{{\hat{\cal
A}}:{\hat{X}}}\rangle\langle{{\hat{\cal
B}}:{\hat{X}}}\rangle}{\langle{{\hat{X}}:{\hat{X}}}\rangle V},$ (60)

where $\delta{\hat{X}}$ is the concentration fluctuation in Eq.(61) with
$\chi=\langle{{\hat{X}}:{\hat{X}}}\rangle$ in Eq.(63). We here use Eq.(62),
the relations below Eq.(70), and Eq.(A14).

5\. Binary mixtures
5.1. Partial quantities and correlations
For binary mixtures, the fluctuating concentration is written as
${\hat{X}}=X+\delta{\hat{X}}$ with $X=n_{2}/n$ and
$\delta{\hat{X}}({\mbox{\boldmath$r$}})=\delta{\hat{X}}_{2}({\mbox{\boldmath$r$}})$
from Eq.(39). From Eqs.(38) and (39) we have

$\delta{\hat{\phi}}={\bar{v}}_{1}\delta{\hat{n}}_{1}+{\bar{v}}_{2}\delta{\hat{n}}_{2},~{}~{}\delta{\hat{X}}=n^{-2}(n_{1}\delta{\hat{n}}_{2}-n_{2}\delta{\hat{n}}_{1}).$
(61)

where $n=n_{1}+n_{2}$. Then, $\delta{\hat{n}}_{1}$ and $\delta{\hat{n}}_{2}$
are written as

$\delta{\hat{n}}_{1}=n_{1}\delta{\hat{\phi}}-{\bar{v}}_{2}n^{2}\delta{\hat{X}},~{}~{}\delta{\hat{n}}_{2}=n_{2}\delta{\hat{\phi}}+{\bar{v}}_{1}n^{2}\delta{\hat{X}}.$
(62)

Since $\langle{{\hat{\phi}}:{\hat{X}}}\rangle=0$ from Eq.(40), $I_{ij}$ are
expressed in terms of the compressibility
$\kappa_{T}=\langle{{\hat{\phi}}:{\hat{\phi}}}\rangle/k_{B}T$ and the
concentration susceptibility $\chi$ with

$\chi=\langle{{\hat{X}}:{\hat{X}}}\rangle=\frac{k_{B}T}{n}{\bigg{(}}\frac{\partial{X}}{\partial{\Delta}}{\bigg{)}}_{T,p}=\frac{n_{1}n_{2}}{n^{3}}+\frac{n_{1}^{2}n_{2}^{2}}{n^{4}}(G_{11}+G_{22}-2G_{12}),$
(63)

where $\Delta=\mu_{2}-\mu_{1}$ is the chemical potential difference (see
Appendix A). We have $\chi\cong{n_{1}n_{2}}/{n^{3}}$ for dilute mixtures or
for mixtures composed of similar components away from the criticality.
However, $\chi$ divereges at the criticality (see Section 5.3). The spinodal
condition $\chi=\infty$ was examined theoretically for highly asymmetric,
hard-sphere mixtures Biben ; Lekker .

In the original papers Buff ; Naim2 ; Naim ; Oc , the partial volumes
${\bar{v}}_{1}$ and ${\bar{v}}_{2}$ are expressed in terms of $G_{ij}$ from
Eq.(14). They are also expressed in terms of $\hat{X}$ and $\chi$ as

$\displaystyle{\bar{v}}_{1}=k_{B}T\kappa_{T}[I_{22}n_{1}-I_{12}n_{2}]/D=\langle{{\hat{n}}_{2}:{\hat{X}}}\rangle/n^{2}\chi,$
$\displaystyle{\bar{v}}_{2}=k_{B}T\kappa_{T}[I_{11}n_{2}-I_{12}n_{1}]/D=-\langle{{\hat{n}}_{1}:{\hat{X}}}\rangle/n^{2}\chi,$
(64)

where $D$ is the determinant of $\\{I_{ij}\\}$ and is calculated from Eq.(62)
as

$D=I_{11}I_{22}-I_{12}^{2}=n^{4}k_{B}T\kappa_{T}\chi.$ (65)

In the dilute limit of the second component, Eq.(64) yields Buff

${\bar{v}}_{2}^{0}=\lim_{n_{2}\to 0}{\bar{v}}_{2}=\lim_{n_{2}\to
0}(k_{B}T\kappa_{T}-G_{12}).$ (66)

In addition, if use is made of ${\hat{n}}={\hat{n}}_{1}+{\hat{n}}_{2}$,
Eq.(65) is rewritten as

$n^{2}k_{B}T\kappa_{T}=\langle{{\hat{n}}:{\hat{n}}}\rangle-\langle{{\hat{n}}:{\hat{X}}}\rangle^{2}/\langle{{\hat{X}}:{\hat{X}}}\rangle.$
(67)

See similar expressions for $\alpha_{p}$ and $C_{p}$ in Eqs.(A15) and (A16).

The partial enthalpies ${\bar{H}}_{i}$ in Eq.(36) are expressed as

$\displaystyle{\bar{H}}_{1}=h{\bar{v}}_{1}-\langle{{\hat{e}}:{\hat{X}}}\rangle\frac{n_{2}}{n^{2}\chi}=h{\bar{v}}_{1}-{\bigg{(}}\frac{\partial{e}}{\partial{X}}{\bigg{)}}_{T,p}\frac{n_{2}}{n^{2}},$
$\displaystyle{\bar{H}}_{2}=h{\bar{v}}_{2}+\langle{{\hat{e}}:{\hat{X}}}\rangle\frac{n_{1}}{n^{2}\chi}=h{\bar{v}}_{2}+{\bigg{(}}\frac{\partial{e}}{\partial{X}}{\bigg{)}}_{T,p}\frac{n_{1}}{n^{2}}.$
(68)

In terms of $J_{i}$ in Eq.(31) we express
$\langle{{\hat{e}}:{\hat{X}}}\rangle$ as

$\langle{{\hat{e}}:{\hat{X}}}\rangle=(J_{2}-J_{1})n_{1}n_{2}/n^{2}.$ (69)

From Eqs.(A14) and (34) we find

${\bigg{(}}\frac{\partial{e}}{\partial{X}}{\bigg{)}}_{T,p}=\langle{{\hat{e}}:{\hat{X}}}\rangle/\chi=({\bar{v}}_{1}{\bar{H}}_{2}-{\bar{v}}_{2}{\bar{H}}_{1})n^{2},$
(70)

which also follows from Eqs.(10) and (11). That is, at fixed $(T,p)$, we have
$dn_{1}=-n^{2}{\bar{v}}_{2}dX$, $dn_{2}=n^{2}{\bar{v}}_{1}dX$, and
$de=dh={\bar{H}}_{1}dn_{1}+{\bar{H}}_{2}dn_{2}$, so $(\partial w/\partial
X)_{T,p}/n^{2}={\bar{v}}_{1}(\partial w/\partial
n_{2})_{T,n_{1}}-{\bar{v}}_{2}(\partial w/\partial n_{1})_{T,n_{2}}$ for any
$w$. From Eqs.(68) and (69) the infinite-dilution limit of ${\bar{H}}_{2}$
becomes

${\bar{H}}_{2}^{0}=\lim_{n_{2}\to 0}{\bar{H}}_{2}=\lim_{n_{2}\to
0}(h{\bar{v}}_{2}+J_{2}-J_{1}).$ (71)

The correlation-function expressions (64) and (68) are convenient for
numerical calculations of ${\bar{v}}_{i}$ and ${\bar{H}}_{i}$. They also
indicate how ${\bar{v}}_{i}$ and ${\bar{H}}_{i}$ behave near the fluid
criticality (see the next subsection). We mention a review on the calculations
of ${\bar{H}}_{i}$ in Ref.Vlugt .

5.2. Scattering intensity and alcohol-water mixtures
In scattering experiments from binary mixtures, the measured intensity in the
long wavelength limit is given by $I=\langle{\hat{\cal C}:\hat{\cal
C}}\rangle$ with $\delta\hat{\cal
C}({\mbox{\boldmath$r$}})=b_{1}\delta{\hat{n}}_{1}+b_{2}\delta{\hat{n}}_{2}+b_{T}\delta{\hat{T}}$,
where $b_{1}$, $b_{2}$, and $b_{T}$ are constants. We express
$\delta{\hat{n}}_{1}$ and $\delta{\hat{n}}_{2}$ in terms of
$\delta{\hat{\phi}}$ and $\delta{\hat{X}}$ using Eq.(62) to find

$I=A_{\phi}^{2}k_{B}T\kappa_{T}+A_{X}^{2}\chi+b_{T}^{2}k_{B}T^{2}/C_{V},$ (72)

where $A_{\phi}={\sum}_{i}b_{i}n_{i}$ and
$A_{X}=n^{2}(b_{2}{\bar{v}}_{1}-b_{1}{\bar{v}}_{2})$. In their theory on light
scattering, Kirkwood and Goldberg Kirk3  set
$A_{\phi}=n(\partial\epsilon/\partial n)_{T,X}$ and
$A_{X}=(\partial\epsilon/\partial X)_{T,p}$ using the optical dielectric
constant $\epsilon$ Kirk3 . Bhatia and Thornton Bhatia  calculated $I$ using
$\delta{\hat{n}}=\delta{\hat{n}}_{1}+\delta{\hat{n}}_{2}$ and
$\delta{\hat{X}}$ with a cross term
$(\propto\langle{{\hat{n}}:{\hat{X}}}\rangle$). In these theories, the last
term in Eq.(72) due to $\delta{\hat{T}}$ was neglected.

In scattering experiments on alcohol-water mixtures in the ambient condition,
the concentration susceptibility $\chi$ in Eq.(63) exhibited a maximum at
intermediate $X$, where $\chi\gtrsim 1/n$ and $G_{11}+G_{22}-2G_{12}\gg 1/n$
Oka1 ; Nishi ; Misawa ; YKoga ; Bagchi1 ; Lep ; Ruck ; Koga , This is because
an amphiphilic species (such as ethanol and tertbutyl-alcohol) tends to
aggregate in liquid water on nanometer scales,

5.3. Critical behaviors in binary mixtures
In near-critical binary mixtures Gri ; Wh ; Cum ; LSengers ; Chimo ; Eck1 ;
Eckert ; Onukibook ; OnukiH , ${\bar{v}}_{i}$ and ${{\bar{H}}_{i}}$ exhibit
complicated crossover behaviors, which has not been studied. Near the gas-
liquid criticality of the primary component 1, ${\bar{v}}_{2}^{0}$ and
${{\bar{H}}_{2}^{0}}$ at infinite dilution $n_{2}\to 0$ grow strongly from
Eqs.(66) and (71). Experimentally, $v_{2}^{0}$ became negatively large near
the solvent gas-liquid critical point Eckert ; Chimo . However,
${\bar{v}}_{2}$ and ${{\bar{H}}_{2}}$ tend to constants for
$n_{2}k_{B}T\kappa_{T}\gtrsim 1$ with increasing $\kappa_{T}$ from Eqs.(64)
and (68).

On approaching a consolute critical line, ${\bar{v}}_{i}$ and
${{\bar{H}}_{i}}$ tend to finite values. This is because
$\langle{\hat{n}_{i}:{\hat{X}}}\rangle$, $\langle{\hat{e}:{\hat{X}}}\rangle$,
and $\chi$ in Eqs.(64) and (68) grow weakly near the non-azeotropic
criticality (with the critical exponent ${\hat{\alpha}}\cong 0.10$), but they
grow strongly near the azeotropic criticality (with the critical exponent
${\hat{\gamma}}\cong 1.24$) Gri ; Onukibook ; Chimo .

6\. Dilute solute in mixture solvents
6.1. Helmholtz free energy up to order $n_{3}^{2}$
We consider a mixture solvent composed of two components $(i=1,2$) and add a
dilute nonionic solute $(i=3)$. Their densities are written as $n_{i}$ with
$n_{3}\ll n_{1}+n_{2}$. The solute has no internal degree of freedom and there
is no chemical reaction. In this section, the solvent composition is
represented by the concentration of the second component,

$X=n_{2}/(n_{1}+n_{2}).$ (73)

In this and next sections, we consider the thermodynamic deviations in the
solvent, $\delta n_{1}$, $\delta n_{2}$,
$\delta\phi={\bar{v}}_{1}\delta{n}_{1}+{\bar{v}}_{2}\delta{n}_{2}$ and
$\delta{X}=n^{-2}(n_{1}\delta{n}_{2}-n_{2}\delta{n}_{1})$ with $n=n_{1}+n_{2}$
(not including $n_{3}$). Note that $\delta{\hat{n}}_{1}$,
$\delta{\hat{n}}_{2}.$$\delta{\hat{\phi}}$, and $\delta{\hat{X}}$ in the
previous sections denote the thermal fluctuations.

For small $n_{3}$ we set up the Helmholtz free energy density up to order
$n_{3}^{2}$ Oka1 ; OnukiH :

$f(\\{n\\},T)=f_{m}(n_{1},n_{2},T)+k_{B}T\big{[}\ln(n_{3}\lambda_{3}^{3})-1+\nu_{3}\big{]}n_{3}+\frac{1}{2}k_{B}Tu_{3}n_{3}^{2},$
(74)

where $f_{m}$ is the solvent free energy density, and $\lambda_{3}(\propto
T^{-1/2})$ is the thermal de Broglie length of the solute. The dimensionless
quantity $\nu_{3}(n_{1},n_{2},T)$ is crucial in this section. It contains an
ideal-gas part Landau-s ,

$\nu_{3}^{0}(T)=-(k_{3}-3/2)\ln(T/\Theta_{3}^{0}),$ (75)

where $2k_{3}-3$ is the vibration-rotation degree of freedom of a solute
particle and $\Theta_{3}^{0}$ is a constant temperature. The solute
contribution to the ideal-gas part of $C_{V}=(\partial e/\partial
T)_{\\{n\\}}$ is then $k_{B}k_{3}n_{3}$ (see Eq.(77)). The difference
$\Delta\nu_{3}=\nu_{3}-\nu_{3}^{0}$ is determined by the solute-solvent
interactions at infinite dilution. The last term in Eq.(74) ($\propto
n_{3}^{2}$) arises from the direct solute-solute interaction under the
influence of the solvent, so $u_{3}$ can depend on $(n_{1},n_{2},T)$. See the
indirect solute-solute interaction from our model in Eq.(93).

From Eq.(74) the solute chemical potential $\mu_{3}=\partial f/\partial
n_{3}$, the energy density $e=-T^{2}\partial(f/T)/\partial T$, and the
pressure $p={\sum}_{i}n_{i}\partial f/\partial n_{i}-f$ are calculated as

$\displaystyle\mu_{3}/k_{B}T=\ln(n_{3}\lambda_{3}^{3})+\nu_{3}+u_{3}n_{3},$
(76) $\displaystyle
e/k_{B}T=e_{m}/k_{B}T+\big{(}\frac{3}{2}-T\nabla_{T}\nu_{3}\big{)}n_{3}-\frac{1}{2}{n_{3}^{2}}T\nabla_{T}{u_{3}}.$
(77) $\displaystyle
p/k_{B}T=p_{m}/k_{B}T+(1+\zeta_{3})n_{3}+\frac{1}{2}n_{3}^{2}\nabla_{n}({n}u_{3}),$
(78)

where $e_{m}(n_{1},n_{2},T)$ is the solvent energy density and
$p_{m}(n_{1},n_{2},T)$ the solvent pressure.

We consider the derivatives of $\nu_{3}(n_{1},n_{2},T)$. Setting
$\nabla_{T}w=(\partial w/\partial T)_{\\{n\\}}$ and $\nabla_{n}w=(\partial
w/\partial n)_{T,X}$ with $n=n_{1}+n_{2}$ for any $w$, we have

$\displaystyle\nu_{31}=(\partial\nu_{3}/\partial
n_{1})_{T,n_{2}},~{}~{}\nu_{32}=(\partial\nu_{3}/\partial n_{2})_{T,n_{1}},$
(79)
$\displaystyle\zeta_{3}=n_{1}{\nu_{31}}+n_{2}{\nu_{32}}=n\nabla_{n}\nu_{3},$
(80) $\displaystyle
g_{3}=n^{2}({\bar{v}}_{1}\nu_{32}-{\bar{v}}_{2}\nu_{31})=(\partial\nu_{3}/\partial
X)_{T,p},$ (81)

In Eq.(81) we can neglect the $n_{3}$-dependence of ${\bar{v}}_{1}$ and
${\bar{v}}_{2}$ and use the first relation in Eq.(12). Here, $\nu_{3i}$,
$\zeta_{3}$, and $g_{3}$ remain finite in the limit $X\to 0$ and 1 and will be
expressed in terms of the KB integrals in Eqs.(98)-(100). For small deviations
of $n_{1}$, $n_{2}$, and $T$, we find

$\delta\nu_{3}=\nu_{31}\delta n_{1}+\nu_{32}\delta
n_{2}+(\nabla_{T}\nu_{3})\delta T=\zeta_{3}\delta\phi+g_{3}\delta
X+(\nabla_{T}\nu_{3})\delta T.$ (82)

The parameter $g_{3}$ represents the asymmetry between the solute-solvent
(3-1) interaction and the solute-cosolvent (3-2) one (see Eq.(100)).

We make some remarks regarding the interaction part
$\Delta\nu_{3}=\nu_{3}-\nu_{3}^{0}$.
(i) For ternary solutions, Ben-Naim Naim3  introduced a solute psudo-chemical
potential, written as $\mu_{s}^{*\ell}$, which is equal to
$k_{B}T\Delta\nu_{3}$ in our notation. He expressed the derivative
$(\partial\mu_{s}^{*\ell}/\partial X)_{T,p}$ in terms of the KB integrals,
which coincides with $g_{3}$ in Eq.(100) multiplied by $k_{B}T$.
(ii) If the solute volume ${\bar{v}}_{3}$ exceeds $1/n$ in a liquid solvent,
each solute particle is in contact with many solvent particles, leading to
$|\Delta\nu_{3}|\gg 1$ for not small preferential solvation. See numerical
calculations of $\Delta\nu_{3}$, $\zeta_{3}$, and $g_{3}$ vs $X$ for various
${\bar{v}}_{3}$ in Fig.3 in Ref.Oka1 . For hard-spheres in ambient liquid
water, Chandler et al. Chan  concluded that $\Delta G(=k_{B}T\Delta\nu_{3})$
is proportional to $R^{2}$ for $R\gtrsim 1$ nm with $R$ being the sphere
radius, which yields $\Delta\nu_{3}\sim 200$ at $R\sim 1$ nm. More discussions
will be given around Eqs.(89) and (97).
(iii) When the solution is phase-separated into liquid and gas regions, the
Ostwald coefficient $L$ ($=$the ratio of the solute densities in the two
phases) Battino  is given by

$L=n_{3}^{\rm liq}/n_{3}^{\rm gas}=\exp[-\Delta\nu_{3}],$ (83)

in the dilute limit of the solute Oka1 ; LSengers ; Ben ; OnukiH . The solute
solubility increases with decreasing $\Delta\nu_{3}$, which is the case of
increasing $X$ for $g_{3}<0$ from Eq.(82).

6.2. Solute partial quantities in terms of $\nu_{3}$ and critical behaviors
Using Eqs.(76)-(78), we can calculate ${\bar{v}}_{3}$ in Eq.(12),
${\bar{H}}_{3}$ in Eq.(17), and $M_{3}$ in Eq.(18) as

$\displaystyle{\bar{v}}_{3}/k_{B}T\kappa_{T}=1+\zeta_{3}+n_{3}\nabla_{n}(nu_{3}),$
(84)
$\displaystyle{\bar{H}}_{3}/k_{B}T={3}/{2}-T\nabla_{T}\nu_{3}+T\alpha_{p}(1+\zeta_{3})-n_{3}T\big{[}\nabla_{T}u_{3}-\alpha_{p}\nabla_{n}(nu_{3})\big{]},$
(85) $\displaystyle
M_{3}/k_{B}T={3}/{2}-T\nabla_{T}\nu_{3}-n_{3}\nabla_{T}u_{3},$ (86)

which are valid up to order $n_{3}$. In the infinite-dilution limit $n_{3}\to
0$, we set

$\displaystyle\kappa_{T}\to\kappa_{T}^{m},~{}~{}\alpha_{p}\to\alpha_{p}^{m},~{}~{}\beta_{V}\to\beta_{V}^{m}.~{}~{}{\bar{v}}_{3}\to{\bar{v}}_{3}^{0},~{}~{}{\bar{H}}_{3}\to{\bar{H}}_{3}^{0},~{}~{}G_{3i}\to
G_{3i}^{0},~{}~{}J_{3}\to J_{3}^{0},$
$\displaystyle{\bar{v}}_{i}\to{\bar{v}}_{i}^{m},~{}~{}I_{ij}\to
I_{ij}^{m},~{}~{}J_{i}\to J_{i}^{m}~{}~{}(i,j=1,2).$ (87)

From Eqs.(84) and (85) we then obtain the infinite-dilution limits of
${\bar{v}}_{3}$ and ${\bar{H}}_{3}$:

$\displaystyle{\bar{v}}_{3}^{0}=\lim_{n_{3}\to
0}{\bar{v}}_{3}=k_{B}T\kappa_{T}^{m}(1+\zeta_{3}),$ (88)
$\displaystyle{\bar{H}}^{0}_{3}=\lim_{n_{3}\to
0}{\bar{H}}_{3}=k_{B}T(3/2-T\nabla_{T}\nu_{3})+T\beta_{V}^{m}{\bar{v}}_{3}^{0}.$
(89)

For relatively large solutes in nearly incompressible solvents with
$n{\bar{v}}_{3}^{0}\gtrsim 1$ and $nk_{B}T\kappa_{T}^{m}\ll 1$, Eq.(88)
indicates $\zeta_{3}\gg 1$ and ${\bar{v}}_{3}^{0}\cong
k_{B}T\kappa_{T}^{m}\zeta_{3}=k_{B}T(\partial\nu_{3}/\partial p)_{T,X}$. For
example, if $n{\bar{v}}_{3}^{0}\sim 5$, we have $\zeta_{3}\sim 100$ in ambient
water-alcohol mixtures Kubota . In addition, $\kappa_{T}$, $\alpha_{p}$,
${\bar{v}}_{i}$, and ${\bar{H}}_{i}$ ($i=1,2$) will be expanded up to order
$n_{3}$ in Appendix B.

Let us consider a near-critical mixture solvent. Here, $\nu_{3}$,
$\nabla_{T}\nu_{3}$, and $\zeta_{3}$ are regular functions of
$(T,n_{1},n_{2})$ even near the criticality, while they are singular as
functions of $(T,p,X)$ near the criticality. As a result, the critical
behaviors of ${\bar{v}}_{3}^{0}$ and ${\bar{H}}_{3}^{0}$ are the same as that
of $\kappa_{T}^{m}$ from Eqs.(88) and (89), so they grow weakly (strongly)
near the non-azeotropic (azeotropic) criticality (see Section 5.2). Note that
many binary mixtures composed of two similar components are nearly azeotropic
at any $X$ Ani ; OnukiL , where the singular part of $\delta{\hat{X}}$ is
small and ${\bar{v}}_{3}^{0}$ and ${{\bar{H}}_{3}^{0}}$ exhibit large critical
enhancement.

6.3. Solute density variances
Using the second relation in Eq.(13) and $\mu_{3}$ in Eq.(76), we obtain

$I^{3i}=\nu_{3i}~{}~{}(i=1,2),~{}~{}I^{33}=1/n_{3}+u_{3},$ (90)

where we neglect $n_{3}\partial u_{3}/\partial n_{i}$ in $I^{3i}$ but retain
$u_{3}$ in $I^{33}$ (to obtain Eq.(93) below). From
${\sum}_{k=1,2}I^{ik}I_{k3}=-I^{3i}I_{33}=-\nu_{3i}I_{33}$ ($i=1,2)$, we have

$I_{3i}/I_{33}=-{\sum}_{k=1,2}\nu_{3k}I^{m}_{ik}~{}~{}(i=1,2).$ (91)

Then,
$1-I^{33}I_{33}={\sum}_{i=1,2}I^{3i}I_{i3}=-I_{33}{\sum}_{i,j=1,2}\nu_{3i}\nu_{3j}I_{ij}^{m}$
from Eqs.(90) and (91), so we obtain

$I_{33}=[I^{33}+{\sum}_{i,j=1,2}I_{ij}^{m}\nu_{3i}\nu_{3j}]^{-1}=n_{3}/[1+n_{3}u_{3}^{\rm
eff}],$ (92)

where $u_{3}^{\rm eff}$ is the effective solute-solute interaction parameter
calculated as Oka1

$\displaystyle u_{3}^{\rm
eff}={u_{3}}-{\sum}_{i,j=1,2}I^{m}_{ij}\nu_{3i}\nu_{3j}$ (93)
$\displaystyle=u_{3}-k_{B}T\kappa_{T}^{m}\zeta_{3}^{2}-\chi g_{3}^{2}.$ (94)

In Eq.(93) the second term stems from the solvent-mediated solute-solute
attraction. In Eq.(94) it is divided into the contribution from the volume-
fraction change ($\propto\zeta_{3}^{2}$) and that from the composition change
($\propto g_{3}^{2}$), where $\zeta_{3}$ and $g_{3}$ are given in Eqs.(80) and
(81) and $\chi$ is defined in Eq.(63) for binary mixtures. From
$I_{33}=n_{3}+n_{3}^{2}G_{33}$ we also have

$G_{33}=-u_{3}^{\rm eff}/(1+n_{3}u_{3}^{\rm
eff}),~{}~{}~{}G_{33}^{0}=\lim_{n_{3}\to 0}G_{33}=-u_{3}^{\rm eff}.$ (95)

If we superimpose small density deviations $\delta n_{i}$ on $n_{i}$ at fixed
$T$, the second order deviation of the displaced free energy density
$f(T,\\{n+\delta n\\})$ is given by ${\sum}_{i,j}I^{ij}\delta n_{i}\delta
n_{j}/2$ from Eq.(13). This expansion also leads to Eqs.(91) and (92) Oka1 .

Let us assume $u_{3}^{\rm eff}<0$ and $|u_{3}^{\rm eff}|\gg 1/n$, which are
realized with increasing the solvent-mediated attraction. In this situation,
we can define the solute spinodal density $n_{3}^{\rm sp}$ by

$I_{33}=n_{3}/(1-n_{3}/n_{3}^{\rm sp}),~{}~{}~{}n_{3}^{\rm sp}=-1/u_{3}^{\rm
eff}=1/G_{33}^{0}\ll n,$ (96)

The solute fluctuation grows as $n_{3}\to n_{3}^{\rm sp}$, indicating solute
aggregation as $n_{3}$ aproaches $n_{3}^{\rm sp}$.

We further make remarks regarding Eq.(94).
(i) The second term in Eq.(94) is rewritten as

$-k_{B}T\kappa_{T}^{m}\zeta_{3}^{2}=-({\bar{v}}_{3}^{0}-k_{B}T\kappa_{T}^{m})^{2}/(k_{B}T\kappa_{T}^{m}),$
(97)

For relatively large solutes with ${\bar{v}}_{3}^{0}\gtrsim 1/n$ in nearly
incompressible solvents, we find $k_{B}T\kappa_{T}^{m}\zeta_{3}^{2}\gtrsim
1/(n^{2}k_{B}T\kappa_{T}^{m}\gg 1/n$ (see the paragraph below Eq.(89)). Thus,
large attraction arises among large solute particles, which is the case even
for one-component solvents or in the limit $X\to 0$ for binary mixtures. This
explains the hydrophobic aggregation in ambient liquid water Chan ; Ben ; Lev
; Oo ; Jiang ; Oka1 ; Bagchi .
(ii) When the two solvent components interact with the solute very
differently, we can have $g_{3}^{2}\chi\gg 1/n$ in Eq.(94) (see Fig.3 in
Ref.Oka1 ). This tendency can appear even for $v_{3}^{0}\sim 1/n$, but is
amplified with increasing $v_{3}^{0}$. In this case, increasing $n_{3}$
results in precipitation of solute-rich domains. This attractive interaction
has been discussed for water-alcohol-hydrophobic solute (or hydrotrope) Hori ;
Zemb1 ; Oka1 .
(iii) For solvent-polymer-colloid solutions, Asakura and Oosawa calculated the
effective potential between colloid particles due to depletion of non-
absorbing polymers near the colloid surfaces Oosawa ; Vrij ; Binder ; Lek1 .
In previous numerical calculations Ben1 ; Roij ; Singh ; Kino , the solute-
solute radial distribution function increased due to the solvent-mediated
attractions if the solute size exceeded the solvent size. In these papers, the
effective solute-solute potential range is on the order of the solute radius.

6.4. Solute-partial quantities in terms of variances
In this subsection, we express the infinite-dilution limits of
($\nu_{3i},\zeta_{3},g_{3},\nabla_{T}\nu_{3})$ and
$({\bar{v}}_{3}^{0},{\bar{H}}_{3}^{0}$) in terms of the $T$-$\mu$ variances.
First, we note $I_{3i}/I_{33}\to n_{i}G_{3i}^{0}$ in Eq.(91), so

$\nu_{3i}=-{\sum}_{j=1,2}n_{j}G_{j3}^{0}\lim_{n_{3}\to 0}I^{ji}~{}~{}(i=1,2),$
(98)

from which $\zeta_{3}$ and $g_{3}$ in Eqs.(80) and (81) are written as

$\displaystyle
k_{B}T\kappa_{T}^{m}\zeta_{3}=-{\sum}_{i=1,2}n_{i}v_{i}^{m}G_{3i}^{0}=-\lim_{n_{3}\to
0}\frac{{\langle{{\hat{n}}_{3}:{\hat{\phi}}}\rangle}}{n_{3}},$ (99)
$\displaystyle
g_{3}=(G^{0}_{31}-G^{0}_{32})\frac{n_{1}n_{2}}{n^{2}\chi}=-\lim_{n_{3}\to
0}\frac{{\langle{{\hat{n}}_{3}:{\hat{X}}}\rangle}}{n_{3}\chi}.$ (100)

Then, Eqs.(88) and (99) give the expression of ${\bar{v}}_{3}^{0}$ originally
derived by Ben Naim Naim3 :

${\bar{v}}_{3}^{0}=k_{B}T\kappa_{T}^{m}-{\sum}_{i=1,2}n_{i}{\bar{v}}_{i}^{m}G_{3i}^{0}.$
(101)

Next, from Eqs.(32) and (86), $M_{3}={\sum}_{i}n_{i}J_{i}I^{i3}$ tends to
$k_{B}T(3/2-T\nabla_{T}\nu_{3})$ as $n_{3}\to 0$. Thus, using Eq.(91), we can
relate $J_{3}^{0}={\lim}_{n_{3}\to 0}J_{3}$ and $\nabla_{T}\nu_{3}$ by

$J_{3}^{0}=k_{B}T(3/2-T\nabla_{T}\nu_{3})-{\sum}_{i=1,2}\nu_{3i}n_{i}J_{i}^{m}.$
(102)

From Eqs.(36), (62), and (89), $\bar{H}_{3}^{0}$ is written as

${\bar{H}}_{3}^{0}=J_{3}^{0}+\lim_{n_{3}\to
0}\big{[}h{\bar{v}}_{3}-\langle{{\hat{e}}:{\hat{\phi}}}\rangle+g_{3}\langle{{\hat{e}}:{\hat{X}}}\rangle\big{]},$
(103)

where $\langle{{\hat{e}}:{\hat{\phi}}}\rangle\to J_{1}^{m}$ and
$\langle{{\hat{e}}:{\hat{X}}}\rangle\to 0$ as $X\to 0$ in agreement with
Eq.(71).

7\. Osmotic equilibrium in ternary fluids
As in the biological situations Pa ; Smith ; Smith1 ; Tim ; Record ; Shimizu ,
we suppose a ternary solution composed of a mixture solvent ($i=1,2$) and a
dilute solute ($i=3$), which is confined in a cell with a fixed volume $V$ in
equilibrium with a large reservoir ($\gg V$). The two regions are separated by
a semipermeable membrane. The solvent can pass through it to acquire common
chemical potentials, $\mu_{1}$ and $\mu_{2}$, in the two regions, while there
is no solute in the reservoir. In the cell, the solute density is increased
from 0 to $n_{3}$ in osmotic equilibrium at fixed $(T,\mu_{1},\mu_{2})$. Then,
all the deviations are functions of $n_{3}$. In the biological papers, the
subscripts 2 and 3 refer to the solute and the cosolvent, respectively.

The osmotic pressure difference $\Pi=p(n_{1},n_{2},n_{3},T)-p_{0}$ arises
across the membrane, where $p_{0}$ is the initial pressure. The reservoir
pressure is unchanged from $p_{0}$. Using the Gibbs-Duhem relation and
$d\mu_{3}=(k_{B}T/I_{33})dn_{3}$, we obtain the differential equation
$d\Pi=n_{3}d\mu_{3}=k_{B}T(1+n_{3}u_{3}^{\rm eff})dn_{3}$. Using $G_{33}^{0}$
in Eq.(95). we find the expansion,

$\Pi/k_{B}T=n_{3}+n_{3}^{2}u_{3}^{\rm eff}/2=n_{3}-n_{3}^{2}G_{33}^{0}/2,$
(104)

which is valid up to order $n_{3}^{2}$ Mc . For one-component solvents,
Eqs.(94) and (95) give $G_{33}^{0}=-U_{33}+k_{B}T\kappa_{T}^{m}\zeta_{3}^{2}$
with Eq.(97) Koga ; Ruck .

We also seek the solute-induced changes in $n_{1}$ and $n_{2}$ in the cell. As
$n_{3}\to 0$, we have

$(\partial n_{i}/\partial
n_{3})_{T,\mu_{1},\mu_{2}}=I_{3i}/I_{33}=n_{i}G_{3i}^{0}~{}~{}(i=1,2).$ (105)

Therefore, $n_{1}$ and $n_{2}$ change from their initial values $n_{i}^{0}$ by

$\Delta n_{i}=n_{i}-n_{i}^{0}=n_{i}G_{3i}^{0}n_{3}~{}~{}(i=1,2).$ (106)

Then, the solvent volume fraction and the solvent concentration change by

$\displaystyle\Delta\phi={\bar{v}}_{1}^{m}\Delta n_{1}+{\bar{v}}_{2}^{m}\Delta
n_{2}=-{\bar{v}}_{3}^{0}n_{3}+k_{B}T\kappa_{T}^{m}n_{3},.$ (107)
$\displaystyle\Delta X=(G_{32}^{0}-G_{31}^{0})(n_{1}n_{2}/n^{2})n_{3}=-\chi
g_{3}n_{3}.$ (108)

where Eq.(14) is used in Eq.(107) and $g_{3}$ in Eq.(108) is given in
Eq.(100). Note that Eq.(107) is consistent with Eq.(11) since the pressure
change is equal to $\Pi\cong k_{B}Tn_{3}$. In Eq.(108), $|\Delta X|$ can much
exceed $n_{3}/n$ for $n|G_{32}^{0}-G_{31}^{0}|\gg 1$ in the presence of
significant preferential solvationm, which is the case for large hydrophobic
solutes in water-alcohol solvents.

In biology Pa ; Smith ; Smith1 ; Tim ; Record ; Shimizu , the degree of
cosolvent adsorption onto (or expulsion from) solute particles is represented
by the preferential binding parameter (PBP) defined by

$\displaystyle\Gamma_{32}={\bigg{(}}\frac{\partial{m_{2}}}{\partial{m_{3}}}{\bigg{)}}_{T,\mu_{1},\mu_{2}}=\frac{n^{2}k_{B}T}{n_{1}n_{3}}{\bigg{(}}\frac{\partial{X}}{\partial{\mu_{3}}}{\bigg{)}}_{T,\mu_{1}.\mu_{2}}$
(109) $\displaystyle=n_{2}(G_{32}^{0}-G_{31}^{0})=-({n^{2}}/{n_{1}})\chi
g_{3}=(n^{2}/{n_{1}}){\Delta X}/{n_{3}},$ (110)

where $m_{i}=n_{i}/n_{1}$ (species molality) and we use
$dm_{2}/m_{2}=dn_{2}/n_{2}-dn_{1}/n_{1}=(n^{2}/n_{1}n_{2})dX$ and $dm_{3}\cong
dn_{3}/n_{1}=n_{3}d\mu_{3}/k_{B}Tn_{1}$. See $g_{3}$ in Eqs.(81) and (100) and
$\Delta X$ in Eq.(108).

As another definition of PBP, the isobaric derivative
$\Gamma_{32}^{\prime}=({\partial m_{2}}/{\partial m_{3}})_{T,p,\mu_{2}}$ has
also been used Tim , where $p$ is fixed instead of $\mu_{1}$. As $n_{3}\to 0$,
we find Record ; Smith1

$\Gamma_{32}^{\prime}=\Gamma_{32}+{n_{2}}(1/{n_{1}}+G_{11}^{m}-G_{12}^{m}).$
(111)

Here, $\Gamma_{32}^{\prime}=w_{2}/w_{3}$ with $w_{i}=(\partial
m_{i}/\partial\mu_{3})_{T.p,\mu_{2}}=(\partial
m_{i}/\partial\mu_{3})_{T.\mu_{1},\mu_{2}}-m_{3}(\partial
m_{i}/\partial\mu_{1})_{T.\mu_{2},\mu_{3}}$. The second term in Eq.(111) is
independent of the solute properties and is negligible for large solutes with
appreciable preferential adsorption, where $n|G_{32}^{0}-G_{31}^{0}|\gg 1$
holds.

Finally, we calculate the solute-induced change in the enthalpy $H$ from its
initial value $H_{0}$ in the cell. At fixed $(V,T,\mu_{1},\mu_{2})$, we have
$dH=V(de+dp)$ with $dp=d\Pi$, so Eqs.(20) and (105) yield the differential
equation,

$dH=V{\sum}_{i}M_{i}(I_{i3}/I_{33})dn_{3}+Vd\Pi=V(J_{3}/k_{B}T+1)d\Pi.$ (112)

Thus, in the linear order in $n_{3}$, the osmotic enthalpy change is written
in terms of $J_{3}^{0}$ as

$\Delta H=H-H_{0}=V(J_{3}^{0}+k_{B}T)n_{3}.$ (113)

We also consider the enthalpy change $(\Delta H)_{r}$ in the reservoir. If the
total solvent numbers in the two regions are fixed, those in the reservoir are
changed by $-V\Delta n_{i}\cong-VI_{3i}$ from Eq.(106). Then, from Eq.(34) we
find

$(\Delta
H)_{r}=-V{\sum}_{i}{\bar{H}}_{i}^{m}I_{3i}+V{\bar{H}}_{3}^{m}I_{33}=V({\bar{H}}_{3}^{0}-k_{B}T^{2}\alpha_{p}^{m}-J_{3}^{0})n_{3}.$
(114)

Therefore, the total enthalpy change is given by $\Delta H+(\Delta
H)_{r}=[{\bar{H}}_{3}^{0}+k_{B}T(1-T\alpha_{p}^{m})]Vn_{3}$.

8\. Summary and remarks
We have presented a fluctuation theory in nonionic multi-component fluids. The
integrals of the correlation functions in the grand-canonical ensemble have
been related to the partial volumes ${\bar{v}}_{i}$, the partial enthalpies
${\bar{H}}_{i}$, and other thermodynamic derivatives.

In Section 2, we have summarized the thermodynamic relations for the partial
quantities. The differential equation (11) is particularly useful, which
reappears in Eq.(47) for the thermal fluctuations. The derivatives of the
energy density $e$ with respect to the number densities $n_{i}$ are written as
$M_{i}$ at fixed $T$ and $R_{i}$ at fixed $p$ in Eqs.(18) and (19) and are
related to ${\bar{H}}_{i}$ in Eq.(22).

In Section 3, we have defined the variance $\langle{{\hat{\cal A}}:{\hat{\cal
B}}}\rangle$ in the grand-canonical ($T$-$\mu$) ensemble in Eq.(28). In
addition to the Kirkwood-Buff integrals $G_{ij}$, we have defined the
variances $n_{i}J_{i}=\langle{{\hat{e}}:{\hat{n}_{i}}}\rangle$ for the energy
density $\hat{e}({\mbox{\boldmath$r$}})$ and the number densities
$\hat{n}_{i}({\mbox{\boldmath$r$}})$. Then, $J_{i}$ are expressed in terms of
$M_{i}$, $R_{i}$, and ${\bar{H}}_{i}$ in Eqs.(32)-(35). We have introduced the
fluctuations of the volume fraction and the concentrations,
$\delta{\hat{\phi}}({\mbox{\boldmath$r$}})$ and
$\delta{\hat{X}}_{i}({\mbox{\boldmath$r$}})$, in Eqs.(38) and (39). The long-
range correlations in the canonical ensemble has also been examined.

In Section 4, we have introduced the fluctuations in the temperature and the
pressure, $\delta{\hat{T}}({\mbox{\boldmath$r$}})$ and
$\delta{\hat{p}}({\mbox{\boldmath$r$}})$, in Eqs.(45) and (46). which are
related to $\delta{\hat{\phi}}({\mbox{\boldmath$r$}})$ by Eq.(47) and satisfy
variance relations in Eq.(48). We have clarified the relationship of our
theory to the Greene-Callen theory Callen1  and the Landau-Lifshitz theory
Landau-s . We have found general variance relations of the stress tensor
${\hat{\Pi}}_{\alpha\beta}({\mbox{\boldmath$r$}})$ with
${\hat{n}}_{i}({\mbox{\boldmath$r$}})$ and ${\hat{e}}({\mbox{\boldmath$r$}})$
in Eq.(51). The long-range correlations in the $T$-$p$ ensemble has been
examined.

In Section 5, we have presented expressions for ${\bar{v}}_{i}$ and
${\bar{H}}_{i}$ for binary mixtures in terms of $J_{i}$ and $I_{ij}$ in
Eqs.(64) and (68). Their critical behavior has also been discussed.

In Section 6, we have treated solutions of a mixture solvent ($i=1,2$) and a
dilute solute ($i=3$), (i) The Helmholtz free energy density $f$ has been
expanded with respect to the solute density $n_{3}$ in Eq.(74), where crucial
is a solute-solvent interaction parameter $\nu_{3}(n_{1},n_{2},T)$. We have
then expressed ${\bar{v}}_{3}$ and ${\bar{H}}_{3}$ in terms of $\nu_{3}$ in
Eqs.(84) and (85). The critical behaviors of the infinite-dilution limits
${\bar{v}}_{3}^{0}$ and ${\bar{H}}_{3}^{0}$ have been discussed. (ii) We have
discussed the solvent-mediated solute-solute attraction in Eqs.(91)-(94),
which is enhanced for large solute sizes and/or significant asymmetry in the
solute-solvent interactions. (iii) We have expressed ${\bar{v}}_{3}$ and
$\bar{H}_{3}$ in terms of the $T$-$\mu$ variances.

In Section 7, we have calculated the solvent density changes $\Delta n_{i}$
and the enthalpy change $\Delta H$ due to addition of a small amount of a
solute in the cell in osmotic equilibrium with a reservoir. In particular, the
solvent concentration change $\Delta X$ and $\Delta H$ have been expressed in
terms of $G_{3i}^{0}={\lim}_{n_{3}\to 0}G_{3i}$ and
$J_{3}^{0}={\lim}_{n_{3}\to 0}J_{3}$.

We make remarks on future problems.
(i) We should construct a fluctuation theory for electrolytes with mixture
solvents, where ion dissociation and association occur particularly for weak
acids and bases in water Ma . Recently, we have examined the ion partial
volumes ${\bar{v}}_{i}$ and the solvent-mediated ion-ion interactions in a
waterlike solvent Oka5 . accounting for the hydration and the steric effect.
We have also calculated the solvent-solvent and solvent-ion correlations due
to the ion solvation, which extend over the inverse Debye wave number
OnukiLong .
(ii) In chemical reactions, heat release (or absorption) and volume changes
are related to ${\bar{H}}_{i}$ and ${\bar{v}}_{i}$ of the reactants and the
products Myers ; Callen ; Landau-s . In biology, protein denaturation has been
studied extensively Pa ; Smith ; Smith1 ; Tim ; Record ; Shimizu . Thus, we
should extend the present theory to include the reaction effects.

Data availability: The data that support the findings of this study are
available within this article.

Appendix A. Correlation functions in the grand-canonical ensemble
For given $(T,\\{\mu\\})$ at a fixed volume, we write the grand-canonical
distribution as Kubo

$P_{\rm g}\propto\exp\Big{[}-\int
d{\mbox{\boldmath$r$}}\Big{(}{\hat{e}}({\mbox{\boldmath$r$}})B-{\sum}_{i}{\hat{n}}_{i}({\mbox{\boldmath$r$}})\xi_{i}\Big{)}\Big{]},$
(A1)

where $B=1/k_{B}T$ and $\xi_{i}=\mu_{i}/k_{B}T$. The average over $P_{\rm g}$
is written as $\langle{\cdots}\rangle$. Then,
$e=\langle{{\hat{e}}({\mbox{\boldmath$r$}})}\rangle$ and
$n_{i}=\langle{{\hat{n}}_{i}({\mbox{\boldmath$r$}})}\rangle$ in the bulk. The
logarithm of the grand-canonical partition function is $\omega V$ with
$\omega=p/k_{B}T$, so $\omega$ obeys Gri ; Kubo ,

$d\omega=-edB+{\sum}_{i}n_{i}d\xi_{i}.$ (A2)

If $B$ and $\xi_{i}$ are slightly changed by $\delta B$ and $\delta\xi_{i}$
homogeneously, $P_{\rm g}$ is changed by

$\delta P_{\rm g}=\int
d{\mbox{\boldmath$r$}}\Big{[}-\delta{\hat{e}}({\mbox{\boldmath$r$}})\delta
B+{\sum}_{i}\delta{\hat{n}}_{i}({\mbox{\boldmath$r$}})\delta\xi_{i}\Big{]}P_{\rm
g}$ (A3)

Then, for any space-dependent fluctuating variable $\hat{\cal
A}({\mbox{\boldmath$r$}})$, its average $\langle{\hat{\cal A}}\rangle$ changes
by

$\delta\langle{\hat{\cal A}}\rangle=-\langle{\hat{\cal
A}:{\hat{e}}}\rangle\delta B+{\sum}_{i}\langle{\hat{\cal
A}:{\hat{n}}_{i}}\rangle\delta\xi_{i},$ (A4)

using the variances in Eq.(28). As a function of $(B,\\{\xi\\}$), the
derivatives of $\langle{\hat{\cal A}}\rangle$ are given by

$\frac{\partial}{\partial B}\langle{\hat{\cal A}}\rangle=-\langle{{\hat{\cal
A}}:{\hat{e}}}\rangle,~{}~{}\frac{\partial}{\partial\xi_{i}}\langle{\hat{\cal
A}}\rangle=\langle{\hat{\cal A}:{\hat{n}}_{i}}\rangle.$ (A5)

Treating $\omega$, $n_{i}$, and $e$ as functions of $(B,\\{\xi\\})$, we have

$\displaystyle\langle{{\hat{n}}_{i}:{\hat{n}}_{j}}\rangle=\partial^{2}\omega/\partial\xi_{i}\partial\xi_{j}=\partial
n_{i}/\partial\xi_{j}=I_{ij},$ (A6)
$\displaystyle\langle{{{\hat{e}}}:{\hat{n}}_{i}}\rangle=-\partial^{2}\omega/\partial\xi_{i}\partial
B=-{\partial n_{i}}/{\partial B}={\partial e}/{\partial\xi_{i}},$ (A7)
$\displaystyle\langle{{{\hat{e}}}:{\hat{e}}}\rangle=\partial^{2}\omega/\partial
B^{2}=-{\partial e}/{\partial B},$ (A8)

where Eq.(A6) leads to Eq.(30). On the other hand, ${\partial
e}/{\partial\xi_{i}}$ in Eq.(A7) is equal to ${\sum}_{j}M_{j}I_{ji}$ from
Eq.(18), leading to Eq.(32). Next, we rewrite Eq.(A8) as

$\langle{{{\hat{e}}}:{\hat{e}}}\rangle=-{\bigg{(}}\frac{\partial{e}}{\partial{B}}{\bigg{)}}_{\\{n\\}}-{\sum}_{i}M_{i}{\bigg{(}}\frac{\partial{n_{i}}}{\partial{B}}{\bigg{)}}_{\\{\xi\\}},$
(A9)

where the first term is $k_{B}T^{2}C_{V}$ and $({\partial n_{i}}/{\partial
B})_{\\{\xi\\}}$ in the second term is
$-\langle{{{\hat{e}}}:{\hat{n}}_{i}}\rangle=-{\sum}_{j}M_{j}I_{ij}$ from
Eq.(A7). Thus, we find Eq.(35).

The local stress tensor ${\hat{\Pi}}_{\alpha\beta}({\mbox{\boldmath$r$}})$ is
microscopically defined with
$\langle{{\hat{\Pi}}_{\alpha\beta}}\rangle=\delta_{\alpha\beta}p$ (see the
sentences above Eq.(51)), so Eq.(A5) gives

$\delta_{\alpha\beta}\frac{\partial p}{\partial
B}=-\langle{{\hat{\Pi}}_{\alpha\beta}:{\hat{e}}}\rangle,~{}~{}\delta_{\alpha\beta}\frac{\partial
p}{\partial\xi_{i}}=\langle{{\hat{\Pi}}_{\alpha\beta}:{\hat{n}}_{i}}\rangle,$
(A10)

which lead to Eq.(51) from Eq.(A2). Generally, for any microscopically defined
$\hat{\cal A}({\mbox{\boldmath$r$}})$, we can determine a variable $\hat{\cal
A}_{\alpha\beta}({\mbox{\boldmath$r$}})$ with
$\langle{{\hat{\Pi}}_{\alpha\beta}:\hat{\cal
A}}\rangle=k_{B}T\langle{\hat{\cal A}_{\alpha\beta}}\rangle$ (see Appendix 1A
in Ref.Onukibook ).

Furthermore, we can rewrite $\delta P_{\rm g}$ in Eq.(A3) as Onukibook

$\delta P_{\rm g}=B\int
d{\mbox{\boldmath$r$}}\Big{[}{n\delta{\hat{\sigma}}{\delta
T}}+\frac{\delta{\hat{n}}}{n}\delta
p+{\sum}_{i}n\delta{\hat{X}}_{i}{\delta\mu_{i}}\Big{]}{P_{\rm g}}.$ (A11)

where $\delta{\hat{X}}_{i}$ are the fluctuating concentrations in Eq.(39), and
$\delta p=s\delta T+{\sum}_{i}n_{i}\delta\mu_{i}$ is the pressure deviation.
We define the fluctuating entropy deviation per particle by

$\displaystyle\delta{\hat{\sigma}}({\mbox{\boldmath$r$}})=[\delta{\hat{e}}-{\sum}_{i}\mu_{i}\delta{\hat{n}}_{i}]/nT-s{\sum}_{i}\delta{\hat{n}}_{i}/n^{2}$
$\displaystyle={\sum}_{i}{\bar{S}}_{i}\delta{\hat{X}}_{i}+C_{p}\delta{\hat{T}}/nT-\alpha_{p}\delta{\hat{p}}/n.$
(A12)

In the second line, $\delta{\hat{T}}$ and $\delta{\hat{p}}$ are the
temperature and pressure fluctuations in Eqs.(45) and (46), so Eq.(48) gives
$\langle{{\hat{T}}:{\hat{\sigma}}}\rangle=k_{B}T/n$ and
$\langle{{\hat{p}}:{\hat{\sigma}}}\rangle=0$. From Eq.(A11) we express the
average deviation $\delta\langle{\hat{\cal A}}\rangle$ in terms of the
thermodynamic deviations $\delta T$, $\delta p$, and $\delta\mu_{i}$ as

$k_{B}T\delta\langle{\hat{\cal A}}\rangle={\langle{\hat{\cal
A}:{\hat{\sigma}}}\rangle}n\delta T+{\langle{\hat{\cal
A}:{\hat{n}}}\rangle}{\delta p}/{n}+{\sum}_{i}{\langle{\hat{\cal
A}:{\hat{X}}_{i}}\rangle}{n\delta\mu_{i}},$ (A13)

which is equivalent to Eq.(A4) in the linear order.

For binary mixtures, we set $\delta{\hat{X}}=\delta{\hat{X}}_{2}$ and
$\delta\Delta=\delta\mu_{2}-\delta\mu_{1}$. Then, the last term in Eq.(A13)
becomes $\langle{\hat{\cal A}:{\hat{X}}}\rangle n\delta\Delta$. At fixed
$(T,p)$, we thus find

$k_{B}T{\bigg{(}}\frac{\partial{\langle{\hat{\cal
A}}\rangle}}{\partial{\Delta}}{\bigg{)}}_{T,p}={\bigg{(}}\frac{\partial{\langle{\hat{\cal
A}}\rangle}}{\partial{X}}{\bigg{)}}_{T,p}{n\chi}={\langle{\hat{\cal
A}:{\hat{X}}}\rangle}{n},$ (A14)

leading to Eqs.(63) and (70) for $\hat{\cal A}={\hat{X}}$ and ${\hat{e}}$.
Since $(\partial w/\partial T)_{p,X}=(\partial w/\partial
T)_{p,\Delta}-(\partial
w/\partial\Delta)_{T,p}\langle{\hat{\sigma}:\hat{X}}\rangle/\chi$ for any $w$,
$\alpha_{p}$ and $C_{p}$ in Eq.(9) are given by Onukibook

$\displaystyle
k_{B}T\alpha_{p}=-\langle{\hat{\sigma}:\hat{n}}\rangle+\langle{\hat{\sigma}:\hat{X}}\rangle\langle{\hat{n}:\hat{X}}\rangle/\chi,$
(A15) $\displaystyle
k_{B}C_{p}/n^{2}=\langle{\hat{\sigma}:\hat{\sigma}}\rangle-\langle{\hat{\sigma}:\hat{X}}\rangle^{2}/\chi.$
(A16)

Here, Eqs.(A15) and (A16) give $\langle{\Delta S\Delta V}\rangle/V$ and
$\langle{(\Delta S)^{2}}\rangle/V$, respectively, in the $T$-$p$ ensemble from
Eq.(60), where $\Delta
S=n\int_{V}d{\mbox{\boldmath$r$}}\delta{\hat{\sigma}}({\mbox{\boldmath$r$}})$.
is the total entropy change. We can also derive $\kappa_{T}$ in Eq.(67) from
Eq.(A13).

Appendix B. Expansions up to order $n_{3}$
In this appendix, we give some additional results for a ternary fluid
containing a dilute solute with density $n_{3}$. From Eqs.(15), (26), and (74)
$\kappa_{T}$, $\beta_{V}$, and $\alpha_{p}=\kappa_{T}\beta_{V}$ are expanded
from their solvent values in Eq.(87) up to order $n_{3}$ in terms of $\nu_{3}$
as

$\displaystyle\kappa_{T}=\kappa_{T}^{m}+\Big{[}2\zeta_{3}+1+{\sum}_{i,j=1,2}\nu_{3ij}n_{i}n_{j}\Big{]}k_{B}T\kappa_{T}^{2}n_{3},$
$\displaystyle\beta_{V}=\beta_{V}^{m}+(1+\zeta_{3}+T\nabla_{T}\nu_{3})k_{B}n_{3},~{}~{}\alpha_{p}/\alpha_{p}^{m}=\kappa_{T}/\kappa_{T}^{m}+\beta_{V}/\beta_{V}^{m}-1,$
(B1)

where $\nu_{3ij}=(\partial^{2}\nu_{3}/\partial n_{i}\partial n_{j})_{T}$
$(i,j=1,2)$.

Next, we present the solvent ${\bar{v}}_{i}$ and ${\bar{H}}_{i}$ ($i=1,2$) up
to order $n_{3}$ from Eqs.(14), (22), (84), (85), and (86) as

$\displaystyle{\bar{v}}_{i}={\bar{v}}_{i}^{m}(1-{\bar{v}}_{3}n_{3})+(\delta_{i2}-X)k_{B}T\kappa_{T}\frac{n_{3}}{n}{\bigg{(}}\frac{\partial{\zeta_{3}}}{\partial{X}}{\bigg{)}}_{T,p},$
(B2)
$\displaystyle{\bar{H}}_{i}={\bar{H}}_{i}^{m}-k_{B}T^{2}(\nabla_{T}\nu_{3})n_{3}+T(\beta_{V}{\bar{v}}_{i}-\beta_{V}^{m}{\bar{v}}_{i}^{m}).$
(B3)

These relations assure ${\sum}_{i}n_{i}{\bar{v}}_{i}=1$ and
${\sum}_{i}n_{i}{\bar{H}}_{i}=h$ up to order $n_{3}$. In the latter we use

$h=\lim_{n_{3}\to
0}h+{\bar{H}}_{3}n_{3}+k_{B}Tn_{3}(1+\zeta_{3})(1-T\alpha_{p}^{m}),$ (B4)

which follows from Eqs.(77) and (85),

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