2.1. Set-up

In this section, we define the physical system under consideration as well as relevant notation. We outline the formalism that will be used to separate the electrons and ions by time scale and relate the partition functions of the subsystems. We then proceed to derive thermodynamic potentials for each subsystem in terms of general interaction potentials. Finally, we outline the procedure for deriving thermal expectation values. The central results of this section are thermodynamic potentials for each species under various constraints. These are listed for electrons with general ion structure factor (2.27), for electrons with equilibrium ions (2.36) and for ions (2.35).

We consider a homogeneous, net-neutral, non-magnetized plasma in volume $V$ consisting of $N_e$ electrons and $N_i = N_e/Z$ ions with charges $q_e = -e$ and $q_i = +Ze$, where for simplicity only a single ion charge state is considered. The mass, position and velocity of particle $j$ of species $s$ are written respectively as $m_s, \boldsymbol r_{s,j}, \boldsymbol v_{s,j}$ and the electron and ion densities are $n_{e} = N_e/V$ and $n_{i} = N_i/V$ respectively. For now, we set the entire system to temperature $T = 1/\beta$. The electrostatic potential at location $\boldsymbol r$ generated by a particle of species $s$ at location $\boldsymbol r_{s,j}$ is denoted by $\varphi _s(\boldsymbol r - \boldsymbol r_{s,j})$ and so the total potential at location $\boldsymbol r$ due to particles of species $s$ is

(2.1)\begin{equation} \phi_s(\boldsymbol r) = \sum_j^{N_s}\varphi_s(\boldsymbol r - \boldsymbol r_{s,j}), \end{equation}

and the energy associated with each species is defined as

(2.2)\begin{equation} E_s = E_{Ks} + E_{\phi s}, \end{equation}

where the kinetic energy is, as usual,

(2.3)\begin{equation} E_{Ks} = \sum_j^{N_s} \frac{1}{2}m_sv_{s,j}^2. \end{equation}

The electrostatic energy is given by

(2.4)\begin{equation} E_{\phi s} = \sum_j^{N_s}q_s \left( \phi_s(\boldsymbol r_{s,j}) - \varphi(0)\right), \end{equation}

where the second term inside the summation subtracts self-energy, the energy of a particle placed at the centre of its own potential well.

There is additionally an inter-species interaction energy $E_\mathrm {int}$, which we can write as the energy of electrons placed in the potential generated by ions, i.e.

(2.5)\begin{equation} E_\mathrm{int} = \sum_j^{N_e} -e\phi_i(\boldsymbol r_{e,j}) . \end{equation}

Having defined each component of the system's microscopic energy, we can proceed with a statistical description. As we will see, partitioning the interaction energy in the two-temperature case is a non-trivial problem. The remainder of this section is devoted primarily to capturing the effect of the interaction energy in a systematic way even out of equilibrium.

2.2. Time scale separation

The partition function offers a statistical description of a system from which various averaged quantities can be determined by taking the appropriate derivatives. For a single-temperature equilibrium plasma, the partition function can straightforwardly be written as

(2.6)\begin{equation} \mathcal{Z} = \int_V {\rm d}X_i \,{\rm d}X_e \exp\left\{-\beta(E_e + E_i + E_\mathrm{int})\right\} . \end{equation}

Here, ${\rm d}X_i, {\rm d}X_e$ are the phase-space integration measures for the ions and electrons respectively, normalized so that ${\rm d}X_s = (1/N_s!)(m_s/2{\rm \pi} \hbar )^{3N_s}\,{\rm d}^{3N_s}r_s\,{\rm d}^{3N_s}v_s$. The subscript $V$ denotes that the spatial integrals are taken over a fixed volume $V$.

When electrons and ions have different temperatures, the standard partition function formalism does not apply. In the absence of interactions, the partition function would be separable into ion and electron parts, and we could simply write the Boltzmann factor for each species using the energy of that species divided by its temperature. To describe the partition of the interaction energy, some authors have attempted to define an effective ‘cross-temperature’ (Seuferling, Vogel & Toepffer Reference Seuferling, Vogel and Toepffer1989; Triola Reference Triola2022). However, there is no consensus on a procedure for choosing this temperature, and no guarantee that a particular choice of effective cross-temperature will yield appropriate physical behaviour. Some models involving a cross-temperature have been shown to correspond well with simulation (Shaffer et al. Reference Shaffer, Tiwari and Baalrud2017); while this validation is useful, a first-principles approach is useful for analysing processes where, for example, energy is added to the system and must be somehow partitioned between species.

To resolve ambiguity associated with the interaction energy, we adopt an approach, similar to that used by Boercker & More (Reference Boercker and More1986), in which the electron dynamics is much faster than the ion dynamics. We allow the electrons to come to equilibrium at temperature $T_e$ within a fixed potential established by the ions. The ions then equilibrate with each other at temperature $T_i$ under constraints imposed by the rapidly established electron shielding. Boercker & More make an ansatz for the way that electron screening affects the ion partition function, which is similar to the Born–Oppenheimer approximation (Essén Reference Essén1977; Dharma-Wardana & Perrot Reference Dharma-Wardana and Perrot1998). Denoting the electron free energy as $A_e$, they apply an additional factor of $\exp \{-A_e/T_i\}$ weighting every ion configuration in the ion partition function (Boercker & More Reference Boercker and More1986). In this way, the electrons provide an effective potential that modifies the $\exp \{-E_i/T_i\}$ term corresponding to bare ion–ion interactions.

In our approach, we incorporate electron response into the ion dynamics using the similar constraint that electrons are in equilibrium at all ion configurations. In other words, the electrons respond rapidly compared with the time scale on which ion configurations change, equilibrating among themselves such that they generate an effective force, which balances exactly the effective force applied by the ions. These effective forces are discussed in more detail below; in brief, the ion subsystem is an equilibrium system that generates a (generalized) force acting on the ions themselves and driving the system toward a thermodynamically favourable state, analogously to the pressure in standard thermodynamics. Positioned in screening clouds around the ions, the electrons also generate a force that acts on the ions, but this force in general drives them toward a different configuration. Our approach imposes that, since the electrons have sufficient time to equilibrate following every change in ion configuration, these forces must be in balance.

We describe our system in a generalized Gibbs ensemble, with generalized displacement $S_{ii}$ (defined below) parameterizing the ion two-point correlation function, and generalized entropic forces $\mathcal {F}_e, \mathcal {F}_i$ acting on this displacement from each subsystem. It is convenient to approach the problem using entropic potentials (Planes & Vives Reference Planes and Vives2002), in part because the resulting expressions make fewer references to species’ temperatures and generalize more readily.

2.3. Definitions

The statistical formalism used in this work is worked out in more detail in Appendix D, starting from basics. Here, we briefly summarize the approach and define quantities that are used in subsequent sections.

Formally, we consider the electrons and ions to be independent subsystems in contact with heat reservoirs of temperatures $T_e$ and $T_i$ respectively. As in many constructions of the canonical ensemble, these baths need not be external to the system, but rather each subsystem can be some subvolume of one species, which is allowed to exchange heat with the rest of that species.

We begin in the canonical ensemble, describing a subsystem of ions in volume $V$ at temperature $T_i = 1/\beta _i$. We define the normalized Fourier-space ion distribution to be

(2.7)\begin{equation} {\tilde{\rho}}_i(\boldsymbol{k}) = \frac{1}{\sqrt{N_i}}\sum_j^{N_i} \exp\left\{{-\text{i}\boldsymbol k\boldsymbol{\cdot} \boldsymbol{r}_{i,j}}\right\}. \end{equation}

Now we use the assumption that statistical quantities in our system are translationally invariant, so that the correlation functions depend only on spatial separation, i.e. if $g_{ab}$ is the two-particle position-space correlation function, then $g_{ab}(\boldsymbol r_2,\boldsymbol r_2) = g_{ab}(\boldsymbol r_2 - \boldsymbol r_1)$. Then the ion–ion structure factor (a normalized Fourier-space two-particle correlation function) depends on only a single $\boldsymbol k$ argument and is defined as

(2.8)\begin{equation} S_{ii}(\boldsymbol k) = |{\tilde{\rho}}_i(\boldsymbol k)|^2 . \end{equation}

We will eventually choose $S_{ii}$ as the thermodynamic variableFootnote ² parameterizing the ion positions, even though the integral form of the partition function (2.9) appears to require knowledge of ${\tilde {\rho }}_i$, which is not uniquely determined by $S_{ii}$. Translational invariance ensures that ${\tilde {\rho }}_i$ will not appear linearly in the thermodynamic potentials. Higher-order correlations (e.g. ${\tilde {\rho }}_i(-\boldsymbol k){\tilde {\rho }}_i(-\boldsymbol q){\tilde {\rho }}_i(\boldsymbol k + \boldsymbol q)$) could appear, but to the order of coupling strength considered in this theory, we will not encounter them.

The ion partition function in the fixed-$S_{ii}$ ensemble is

(2.9)\begin{equation} \mathcal{Z}_i(\beta_i, V, S_{ii}) = \int_{V,S_{ii}} {\rm d}X_i \exp\{{-\beta_i E_i(X_i)} \}, \end{equation}

where the subscripts $V, S_{ii}$ indicate that we only integrate over regions of ion phase space that are consistent with the known volume $V$ and known structure factor $S_{ii}$. We note that the ion phase space is represented by the positions of the $N_i$ discrete particles and so, in general, different values of $S_{ii}$ occupy different phase-space volumes in the integral over ${\rm d}X_i$. The configurational entropy associated with fixing $S_{ii}$ therefore varies with the choice of $S_{ii}$.

We now consider a subsystem of electrons in the same volume at inverse temperature $\beta _e$. Because we have grouped the interaction energy $E_\mathrm {int}$ into the electron subsystem, the total energy of the electrons depends on the configurations of both ions and electrons. However, we will choose a form for the interaction energy in which the dependence on ion configuration enters only through ${\tilde {\rho }}_i$. Therefore, we can write the electron partition function $\mathcal {Z}_e$ as

(2.10)\begin{equation} \mathcal{Z}_e(\beta_e, V, S_{ii}) = \int_{V} {\rm d}X_e \exp\{{-\beta_e (E_e(X_e) + E_\mathrm{int}(X_e, {\tilde{\rho}}_i))}\} , \end{equation}

where the ${\tilde {\rho }}_i$ used as a parameter of $E_\mathrm {int}$ is any Fourier-space distribution ${\tilde {\rho }}_i$ compatible with the fixed $S_{ii}$. By the argument above, the phase on ${\tilde {\rho }}_i$ is irrelevant as long as we are considering only two-particle correlations, as we do here. We can then write the corresponding entropic Massieu potential as

(2.11)\begin{equation} \varPhi_e(\beta_e, V, S_{ii}) = \ln \mathcal{Z}_e . \end{equation}

See Appendix D for more detailed discussion of the thermodynamic potentials used in this work. In short, the Massieu potential is an entropic analogue to the Helmholtz free energy, $A_e = -T_e \varPhi _e$, which is convenient in this two-temperature system because its use means that fewer species-dependent temperature factors appear in our thermodynamic relations and that entropic quantities for the subsystems add without relative weighting factors.

With these expressions, we could describe the thermodynamics of electrons and ions in some volume if we knew the ion distribution in that volume. In general, however, we do not know $S_{ii}$ a priori; rather, we want to solve for its thermal average under the constraint of some known extensive quantity. The natural quantity is the generalized force applied to the electrons by the ions, which we can find by requiring that the electrons come to equilibrium separately for every ion microstate. This assumption is discussed in detail through examples in Appendix E. We transform to an ensemble in which we apply a constraint on this force and allow $S_{ii}$ to fluctuate.

For the ion subsystem, we consider $S_{ii}(\boldsymbol k)$ at all $\boldsymbol k$, and define the entropic force as the conjugate variable $(\hat {V} \mathcal {F}_i)$, where $\hat {V} = V/(2{\rm \pi} )^3$ is the volume scaled by a factor that is convenient with the present Fourier transform convention. The quantity $\mathcal {F}_i$ (without the factor of volume) is the one that we will most often use. In the canonical ensemble, we would impose that the ion subsystem be in contact with some external reservoir, enforcing that the reservoir shares the same $S_{ii}$ as the ions but allowing $\mathcal {F}_i$ to fluctuate.

If instead $\mathcal {F}_i$ is known in the reservoir (and therefore $S_{ii}$ is of course allowed to fluctuate), then we can write the partition function by the same procedure as in (D15), giving

(2.12)\begin{equation} \zeta_i(\beta_i, V, (\hat{V} \mathcal{F}_i)) = \int_{V} {\rm d}X_i \exp\{{-\beta_i E_i(X_i) - \mathcal{F}_i \cdot S_{ii}}\} , \end{equation}

where as shorthand, we have defined a dot product over functions of $\boldsymbol k$ such that

(2.13)\begin{equation} g \cdot h \doteq V\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} g^*(\boldsymbol k) h(\boldsymbol k) . \end{equation}

In order to write this change of ensembles in the form of a Legendre transform, we define the average $\langle q\rangle$ over ion configurations of some quantity $q$ in the standard way as

(2.14)\begin{equation} \langle q \rangle = \int_V {\rm d}X_i q(X_i) p_i(X_i), \end{equation}

where the ion probability density is

(2.15)\begin{equation} p_i(X_i) = \zeta_i^{{-}1} \exp\{{-\beta_i E_i(X_i) - \mathcal{F}_i\cdot S_{ii}(X_i)}\} . \end{equation}

The entropic potential $\varPsi _i = \ln \zeta _i$ in the fixed $\mathcal {F}_i$ ensemble is then

(2.16)\begin{equation} \varPsi_i (\beta_i, V, (\hat{V} \mathcal{F}_i)) = \varPhi_i(\beta_i, V, \langle S_{ii} \rangle) - \mathcal{F}_i \cdot \langle S_{ii} \rangle . \end{equation}

We can do the same for the electron subsystem to write an electron partition function in terms of the entropic force $\mathcal {F}_e$ acting on the electrons,Footnote ³ which gives

(2.17)\begin{equation} \zeta_e(\beta_e, V, (\hat{V} \mathcal{F}_e)) = \int_{V} {\rm d}X_e \exp\{{-\beta_e (E_e(X_e) + E_x(X_e)) - \mathcal{F}_e \cdot S_{ii}}\} , \end{equation}

while the electron Planck potential is

(2.18)\begin{equation} \varPsi_e (\beta_e, V, (\hat{V} \mathcal{F}_e)) = \varPhi_e(\beta_e, V, \langle S_{ii} \rangle) - \mathcal{F}_e \cdot \langle S_{ii} \rangle . \end{equation}

Now, in order to evaluate the ion partition function, we need a prescription for fixing $\mathcal {F}_i$. We start by finding the electron response to a given ion distribution, which is the entropic force $\mathcal {F}_e$, found from the electron Massieu potential using

(2.19)\begin{equation} (\hat{V} \mathcal{F}_e) = \left(\frac{\delta \varPhi_e}{\delta S_{ii}}\right)_{\beta_e, V} . \end{equation}

We now say that the reservoir with which the ions are interacting is the electron subsystem. The force from the reservoir can therefore be identified with the force $F_e$ due to the electrons. We apply this condition at all wavevectors $\boldsymbol q$, i.e. $F_e(\boldsymbol q) + F_i(\boldsymbol q) = 0$. This constraint is discussed in more detail through a toy model in Appendix E. Defining the temperature ratio $\tau = T_e/T_i$ for convenience and noting that the force $F_R$ applied by a reservoir of temperature $T_R$ is related to the entropic forces by $F_R = T_R\mathcal {F}_R$, the equilibrium condition becomes

(2.20)\begin{equation} \left.\begin{gathered} 0 =T_e\mathcal{F}_e + T_i\mathcal{F}_i,\\ \mathcal{F}_i ={-} \tau \frac{1}{\hat{V}}\left(\frac{\delta \varPhi_e}{\delta S_{ii}}\right)_{\beta_e, V} . \end{gathered}\right\} \end{equation}

It is understood that everywhere $\mathcal {F}_i$ appears in future expressions, we insert the entropic force as determined by the equilibrium constraint (2.20). We note that this constraint relies only on the other parameters of the two subsystems, which in this case are $\{\beta _i, V, \beta _e\}$. Therefore, we can write a form for $\varPsi _i$ that depends only on volume and on the temperatures of the two species, where electron equilibrium for all ion configurations is implicitly assumed. We write this version of the ion Planck potential as

(2.21)\begin{equation} \hat{\varPsi}_i(\beta_e, \beta_i, V) = \ln \zeta_i, \end{equation}

where the hat on $\hat {\varPsi }$ is meant as a reminder that the arguments $\beta _e, \beta _i, V$ are no longer the natural variables of the Planck potential, and therefore the expected thermodynamic relations may no longer hold when taking partial derivatives of $\hat {\varPsi }_i$. For instance, if $\mathcal {F}_i$ depends on $\beta _i$ (which we will find later in the case of interest), we have in general that

(2.22)\begin{equation} \left(\frac{\partial \hat{\varPsi}_i(\beta_e, \beta_i, V)}{\partial \beta_i}\right)_{\beta_e, V} \neq \left(\frac{\partial \varPsi_i(\beta_i, V, (\hat{V} \mathcal{F}_i))}{\partial \beta_i}\right)_{\mathcal{F}_i, V}, \end{equation}

where the derivative of $\varPsi _i(\beta _i, V, \mathcal {F}_i)$ is taken at constant $(\hat {V} \mathcal {F}_i)$, and then the (generally $\beta _i$-dependent) expression for $(\hat {V} \mathcal {F}_i)$ is substituted in the final step.

2.4. Electron subsystem

With this formalism in place, we can apply it to describe a plasma of electrons and ions with a large separation of dynamical time scales.

We first evaluate the electron energy and partition function. We work in Fourier space with the electrostatic potential

(2.23)\begin{equation} \tilde{\phi}(\boldsymbol k) = \int {\rm d}^3r\, \phi(\boldsymbol r)\exp({-{\rm i}\boldsymbol k\boldsymbol{\cdot} \boldsymbol r}), \end{equation}

and normalized Fourier-space electron distribution

(2.24)\begin{equation} {\tilde{\rho}}_e(\boldsymbol k) = \frac{1}{\sqrt{N_e}}\sum_j^{N_e} \exp({-{\rm i}\boldsymbol k \boldsymbol{\cdot} \boldsymbol r_{e,j}}) . \end{equation}

For generality, and notational simplicity, we will describe the electrostatic potential in terms of Green's functions $C_{ab}(\boldsymbol k)$ acting on the particle distributions; these functions can be specified at the end of the calculation, allowing the derivation to apply for a variety of effective potentials. Gathering temperature, volume and charge factors in the Green's function definitions for convenience, the potential seen by the electrons is defined to be

(2.25)\begin{equation} \tilde{\phi}(\boldsymbol k) ={-}V\frac{T_e}{e} \frac{1}{\sqrt{N_e}} \left[C_{ee}(\boldsymbol k){\tilde{\rho}}_e(\boldsymbol k) + C_{ei}(\boldsymbol k){\tilde{\rho}}_i(\boldsymbol k)\right] . \end{equation}

This definition allows the potential energy to be written in simple form as

(2.26)\begin{equation} E_{\phi e} = T_eC_{ee}{\tilde{\rho}}_e\cdot{\tilde{\rho}}_e + T_eC_{ei}{\tilde{\rho}}_i\cdot{\tilde{\rho}}_e - T_eC_{ee} \cdot 1. \end{equation}

The final term in this equation, included to remove the contribution from self-interactions, has a possibly counterintuitive form but is discussed further in Appendix B.1.

The calculation of the electron partition function using this expression is presented in detail in Appendix B.2. Here, we present only the results that will be used elsewhere in the work.

It follows from (B23) that the Massieu potential for the electrons is

(2.27)\begin{align} \varPhi_e & ={-}\frac{V}{2} \int \frac{{\rm d}^3k}{(2{\rm \pi})^3} \left[ \ln\left(2C_{ee} + 1 \right) - \frac{C_{ei}^2}{2 C_{ee}+1}S_{ii} - 2C_{ee} \right]\nonumber\\ & \quad - \ln N_e! + N_e\ln\left(\left(\frac{m_eT_e}{2{\rm \pi} \hbar^2}\right)^{3/2}V\right) . \end{align}

The entropic force applied by the electrons to systems with which they are in force balance is then

(2.28)\begin{equation} \mathcal{F}_e(\boldsymbol k) = \frac{\frac{1}{2}C_{ei}^2(\boldsymbol k)}{2C_{ee}(\boldsymbol k) + 1} . \end{equation}

The condition of force balance (2.20) requires that

(2.29)\begin{equation} \mathcal{F}_i(\boldsymbol k) ={-}\tau\frac{\frac{1}{2} C_{ei}^2(\boldsymbol k)}{2 C_{ee}(\boldsymbol k) + 1} . \end{equation}

We can now use this expression for $\mathcal {F}_i$ to write the ion partition function in the generalized Gibbs ensemble, in which we impose electron equilibrium for all ion configurations.

2.5. Ion subsystem

Now using the definitions of $\zeta _e, \zeta _i$ from (2.17) and (2.12), we can write the partition functions for each subsystem. The electrons are straightforward because the additional force term can be taken outside of the integral over electron configurations, meaning $\zeta _e(\beta _e, V, (\hat {V} \mathcal {F}_e)) = \exp ({-\mathcal {F}_e\cdot \langle S_{ii} \rangle })\mathcal {Z}_e(\beta _e, V, \langle S_{ii} \rangle )$. The ion partition function that we are interested in finding is

(2.30)\begin{equation} \zeta_i = \int {\rm d}X_i \exp\{{-E_i/T_i - \mathcal{F}_i\cdot S_{ii}}\} . \end{equation}

As we did for the electrons, we will write the ion–ion interaction potential in a general form using Green's functions, so that the ion potential energy is

(2.31)\begin{equation} E_{\phi i} = T_iC_{ii}{\tilde{\rho}}_i \cdot {\tilde{\rho}}_i - T_iC_{ii} \cdot 1. \end{equation}

Then the ion partition function in the generalized Gibbs ensemble is

(2.32)\begin{equation} \zeta_i = \mathcal{Z}_{Ki}\mathcal{Z}_{\phi i}^\mathrm{self}\int {\rm d}^{3N_i}r_{i,j} \exp\left\{{-}C_{ii}\cdot S_{ii} - \mathcal{F}_i \cdot S_{ii}\right\}, \end{equation}

where the self-energy correction has been factored out into $\mathcal {Z}_{\phi i}^\mathrm {self}$. This partition function can also be written as

(2.33)\begin{equation} \zeta_i = \mathcal{Z}_{Ki}\mathcal{Z}_{\phi i}^\mathrm{self}\int {\rm d}^{3N_i}r_{i,j} \exp\left\{{-}C_{ii}^\prime{\tilde{\rho}}_i\cdot{\tilde{\rho}}_i\right\}, \end{equation}

where $C_{ii}^\prime = C_{ii} + \mathcal {F}_i$ is an effective ion–ion interaction Green's function. This integral has the same form as (B10), if we make the symbolic substitutions $C_{ii} \rightarrow C_{ii}^\prime, C_{ei} \rightarrow 0$, meaning that we can immediately write the result of the integral over ion positions (cf. Appendix B.2) as

(2.34)\begin{equation} \zeta_{\phi i} = \exp\left\{-\frac{V}{2} \int \frac{{\rm d}^3k}{(2{\rm \pi})^3} \left[\ln\left(2C_{ii}^\prime + 1 \right)- 2C_{ii} \right]\right\}, \end{equation}

and so the full ion Planck potential becomes

(2.35)\begin{align} \varPsi_{i}(\beta_i, V, \mathcal{F}_i) & ={-}\frac{V}{2}\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} \left[\ln(2C_{ii} + 2\mathcal{F}_i + 1) - 2C_{ii}\right]\nonumber\\ & \quad - \ln N_i! + N_i\ln\left(\left(\frac{m_iT_i}{2{\rm \pi}\hbar^2}\right)^{3/2}V\right) . \end{align}

Physically, this means that the ions interact as a OCP under some effective potential, which is a combination of the bare Coulomb potential and a term due to electron screening.

For completeness, we can also transform the electron Massieu potential to the new ensemble, giving the electron Planck potential explicitly as

(2.36)\begin{align} \varPsi_e(\beta_e, V, \mathcal{F}_e) & ={-} \frac{V}{2}\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} \left[ \ln\left(2C_{ee} + 1 \right) - 2C_{ee} - \left(\frac{C_{ei}^2}{2 C_{ee}+1} - 2\mathcal{F}_e\right)\langle S_{ii} \rangle \right]\nonumber\\ & \quad - \ln N_e! + N_e\ln\left(\left(\frac{m_eT_e}{2{\rm \pi}\hbar^2}\right)^{3/2}V\right). \end{align}

In the system of interest, kinetic degrees of freedom are fully separable and so the kinetic part of the partition function always yields the ideal-gas result in our classical model. We will therefore sometimes only work with the electrostatic components of the thermodynamic potentials, $\varPhi _{\phi e}, \varPsi _{\phi e}, \varPsi _{\phi i}$, corresponding to only the electrostatic energy. These are defined explicitly in Appendix B.1.

We could in principle now substitute the electron force given by (2.28) and get explicit expressions for the Planck potentials $\hat {\varPsi }_e, \hat {\varPsi }_i$ in terms of only the externally imposed variables $\beta _e, \beta _i, V$. However, in the generalized Gibbs ensemble that we have set up, differentiating the partition function should be done while holding the force fixed; it is therefore most useful to keep the potential in this expanded form for the following steps, and apply our knowledge of the force only in the final step.

The equations for the Planck potentials $\varPsi _e$ (2.36) and $\varPsi _i$ (2.35) serve as equations of state for the subsystems of a two-temperature plasma. Together, they comprise a quasi-EOS. It bears repeating that, within the regime of applicability of our assumptions, these expressions could be applied to arbitrary two-particle interactions. In § 3 we will specialize to a specific physically relevant case, but first we discuss the procedure for generating thermal averages from these equations of state.

2.6. Thermal expectation values

Using the Planck potentials for each species, we can calculate various quantities at thermal equilibrium. It will be of interest to calculate some quantities when the electrons are at equilibrium but the ion structure factor is held fixed, as well as when both subsystems are at equilibrium. We have already defined $S_{ii}$ in (2.8). We can define electron–electron and electron–ion structure factors as well; since in this work we are only interested in time scales longer than the electron equilibration time, quantities will always be averaged over electron configurations. The new structure factors are therefore defined as

(2.37)\begin{equation} \left.\begin{gathered} S_{ee}(\boldsymbol q) = \int {\rm d}X_e p_e(X_e) {\tilde{\rho}}_e^*(\boldsymbol q){\tilde{\rho}}_e(\boldsymbol q)\\ S_{ie}(\boldsymbol q) = \int {\rm d}X_e p_e(X_e) {\tilde{\rho}}_e^*(\boldsymbol q){\tilde{\rho}}_i(\boldsymbol q) \end{gathered}\right\}, \end{equation}

where the electron probability density is

(2.38)\begin{equation} p_e = \mathcal{Z}_e^{{-}1} \exp\{{-\beta_e(E_e + E_\mathrm{int})}\}, \end{equation}

in the canonical ensemble. Since $p_e$ depends on $S_{ii}$, so do $S_{ee}$ and $S_{ie}$. When the ions are also at equilibrium, we can substitute the equilibrium ion–ion structure factor $\langle S_{ii} \rangle$ in order to find the other equilibrium structure factors $\langle S_{ee} \rangle, \langle S_{ie} \rangle$.

We will start with the ion structure factor, which can be found by differentiating the ion Planck potential with respect to $C_{ii}$ to be

(2.39)\begin{equation} \langle S_{ii}(\boldsymbol q) \rangle = 1 - \frac{1}{\hat{V}} \left(\frac{\delta \varPsi_i}{\delta C_{ii}(\boldsymbol q)}\right)_{V,\mathcal{F}_i}, \end{equation}

where we denote by the subscript that the generalized force should be held constant because $(\hat {V} \mathcal {F}_i)$ constitutes an independent thermodynamic variable, which should be unaffected by the differentiation even if the final formulas for this quantity depend on $C_{ii}$. The factor $1/\hat {V} = (2{\rm \pi} )^3/V$ simply removes the prefactor on the integral that appears in the expression for $\varPsi _i$. Some manipulation yields

(2.40)\begin{equation} \langle S_{ii}(\boldsymbol q) \rangle = \frac{1}{1 + 2 C_{ii}(\boldsymbol q) - \tau \dfrac{C_{ei}^2(\boldsymbol q)}{2 C_{ee}(\boldsymbol q)+1}} . \end{equation}

Here, we have used $\mathcal {F}_e = {\frac {1}{2} C_{ei}^2}/({2 C_{ee}+1})$ from the derivation of the entropic force in (2.28). The ion–electron structure factor is similarly given by

(2.41)\begin{equation} S_{ie}(\boldsymbol q) ={-}\frac{1}{\hat{V}}\left(\frac{\delta \varPhi_e}{\delta C_{ei}(\boldsymbol q)}\right)_{S_{ii}, V} , \end{equation}

which simplifies to

(2.42)\begin{equation} S_{ie}(\boldsymbol q) ={-}\frac{C_{ei}(\boldsymbol q)}{2C_{ee}(\boldsymbol q)+1} S_{ii}(\boldsymbol q) . \end{equation}

Finally, for the electrons, the structure factor is given by

(2.43)\begin{equation} S_{ee}(\boldsymbol q) = 1 - \frac{1}{\hat{V}} \left(\frac{\delta \varPhi_e}{\delta C_{ee}(\boldsymbol q)}\right)_{S_{ii}, V}, \end{equation}

and therefore

(2.44)\begin{equation} S_{ee}(\boldsymbol q) = \frac{1}{2C_{ee}(\boldsymbol q) + 1} + \frac{C_{ei}^2(\boldsymbol q)}{(2C_{ee}(\boldsymbol q)+1)^2} S_{ii}(\boldsymbol q) . \end{equation}

Using these structure factors, we could obtain the potential energy directly by integrating, but it is informative to derive it instead from the partition function by differentiating the Planck potential with respect to the temperature. The energy of each species is given by

(2.45)\begin{equation} U_a ={-}\left(\frac{\partial\varPsi_a}{\partial \beta_a}\right)_{V, \mathcal{F}_a} . \end{equation}

The kinetic part gives the usual ideal-gas result. For the electrostatic part, using the fact that we absorbed the temperatures into the definitions of the Green's functions, and so each temperature appears only as $\beta _a$ in the corresponding $C_{ab}$, we can differentiate with respect to each Green's function. The potential components of each species’ energy are then

(2.46)\begin{equation} \left.\begin{gathered} U_{\phi i} ={-}T_i\frac{1}{\hat{V}}C_{ii}\cdot\left(\frac{\delta\varPsi_{i}}{\delta C_{ii}}\right)_{V, \mathcal{F}_i}\\ U_{\phi e} ={-} T_e\frac{1}{\hat{V}} C_{ee}\cdot\left(\frac{\delta\varPsi_{e}}{\delta C_{ee}} \right)_{V, \mathcal{F}_e} - T_e\frac{1}{\hat{V}}C_{ei}\cdot\left(\frac{\delta\varPsi_e}{\delta C_{ei}} \right)_{V, \mathcal{F}_e} . \end{gathered}\right\} \end{equation}

These derivatives are the same ones that yielded the structure factors above, meaning that the energy can also be written as

(2.47)\begin{equation} U = \tfrac{3}{2}N_eT_e + \tfrac{3}{2}N_iT_i + T_e C_{ee}\cdot (S_{ee} - 1) + T_e C_{ei}\cdot S_{ie} + T_iC_{ii}\cdot (S_{ii} - 1) . \end{equation}

This is exactly the prescription for integrating the structure factor that we would otherwise have used to calculate the energy directly from known two-particle correlations. This is a useful check on the formalism adopted here, in that the two-time scale equilibrium procedure has not left artefacts in the thermodynamic state functions that we get from $\zeta _e$ and $\zeta _i$. The partition function approach adopted here also includes a prescription for dividing the potential energy between electrons and ions, which would not be uniquely specified when simply integrating the structure factor and potential to find the total energy. The electron electrostatic energy, given as a function of the structure factor for greater generality, is

(2.48)\begin{align} U_{\phi e}(T_e; S_{ii}) & = T_e V\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} C_{ee} \left( \frac{1}{2 C_{ee} + 1} - 1 + \frac{C_{ei}^2}{(2C_{ee}+1)^2}S_{ii} \right)\nonumber\\ & \quad - T_e V\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} C_{ei} \frac{C_{ei}}{2C_{ee}+1}S_{ii} . \end{align}

When the ions are at equilibrium, we can substitute the $S_{ii}$ derived above, and the electron electrostatic energy reduces to

(2.49)\begin{align} U_{\phi e}(T_e, T_i) & = T_e V\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} \left( - \frac{2C_{ee}^2 }{2 C_{ee} + 1} + \frac{C_{ee}}{(2C_{ee}+1)}\frac{C_{ei}^2}{(2C_{ee}+1)(1 + 2 C_{ii}) - \tau C_{ei}^2} \right)\nonumber\\ & \quad- T_eV\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} \frac{C_{ei}^2}{(2C_{ee}+1)(1 + 2 C_{ii}) - \tau C_{ei}^2} . \end{align}

Finally, the ion electrostatic energy is

(2.50)\begin{equation} U_{\phi i}(T_e, T_i) = T_i V\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} C_{ii} \left(\frac{1}{1 + 2 C_{ii} - \tau \dfrac{C_{ei}^2}{2 C_{ee}+1}} - 1\right). \end{equation}

We define the entropy in this work using the Gibbs definition $\mathcal {S} = -\int \textrm {d}X_s p_s\ln p_s$ where $p_s$ is the phase-space probability density of species $s$. The ion entropy, described by the generalized Gibbs ensemble, can then be written as

(2.51)\begin{equation} \left.\begin{gathered} \mathcal{S}_i = \zeta_i^{{-}1}\int {\rm d}X_i \left(\ln\zeta_i + \beta_iE_i + \mathcal{F}_i\cdot S_{ii} \right) \exp({-\beta_iE_i - \mathcal{F}_i \cdot S_{ii}})\\ \mathcal{S}_i = \varPsi_i + \beta_i U_i + \mathcal{F}_i\cdot\langle S_{ii} \rangle . \end{gathered}\right\} \end{equation}

If system is constrained by known $S_{ii}$ rather than by the condition of force balance, then we can still compute an entropy for the electrons, but we should use the canonical ensemble instead of the generalized Gibbs ensemble. In this case the electron entropy is given by

(2.52)\begin{equation} \mathcal{S}_e = \varPhi_e + \beta_e (U_e + U_x). \end{equation}

This is equivalent, when the ions are at equilibrium, to the expression ${\mathcal {S}_e = \varPsi _e + \beta _e(U_e + U_x) + \mathcal {F}_e\cdot \langle S_{ii} \rangle }$ found from the generalized Gibbs ensemble.

3.1. Coulomb potentials

Having obtained general expressions for the thermodynamic potentials of electrons and ions, we can specialize to a case of interest. This requires specifying the functional form of the electron–electron, ion–ion and electron–ion interactions. In this section, we specify such forms and calculate explicit expressions for the resulting thermal averages.

To describe a classical plasma consisting of electrons and ions, we choose Green's functions corresponding to the Coulomb potential. It is necessary to include the factors of temperature, charge, and particle number that we grouped into the Green's functions to simplify notation in (2.25). Including a factor $\frac {1}{2}$ to avoid overcounting within the same species, the Coulomb Green's functions are

(3.1)\begin{equation} \left.\begin{gathered} C_{ee}(\boldsymbol k) = \frac{1}{2}N_e\frac{4{\rm \pi} e^2}{VT_e}\frac{1}{k^2} = \frac{1}{2}\frac{\kappa^2}{k^2} ,\\ C_{ei}(\boldsymbol k) ={-}\sqrt{N_eN_i}\frac{4{\rm \pi} Z e^2}{VT_e}\frac{1}{k^2} ={-}\frac{\sqrt{Z}\kappa^2}{k^2} ,\\ C_{ii}(\boldsymbol k) = \frac{1}{2}N_e\frac{4{\rm \pi} Z^2 e^2}{VT_i}\frac{1}{k^2} = \frac{1}{2}\frac{\chi^2}{k^2}, \end{gathered}\right\} \end{equation}

where $k = |\boldsymbol k|$ and we have defined the inverse screening lengths for electrons and ions, respectively, as $\kappa = \sqrt {4{\rm \pi} e^2 n_{e}/T_e}$ and $\chi = \sqrt {4{\rm \pi} Z^2e^2 n_{i}/T_i}$. These quantities are inversely proportional to single-species Debye lengths for electrons and ions. We can likewise define the total inverse screening length in the usual way as $k_D^2 = \kappa ^2 + \chi ^2$. In these variables, the entropic force from the electron subsystem given by (2.28) is then

(3.2)\begin{equation} \mathcal{F}_e = \frac{1}{2} \frac{1}{k^2}\frac{Z\kappa^4}{\kappa^2 + k^2} . \end{equation}

Notably, the effective Green's function $C_{ii}^\prime = C_{ii} + \mathcal {F}_i$ for ion–ion interactions then simplifies to

(3.3)\begin{equation} C_{ii}^\prime = \frac{1}{2} \frac{\chi^2}{k^2 + \kappa^2}, \end{equation}

which is a Yukawa potential of screening parameter $\kappa$. The ions can therefore be treated as an isolated gas interacting through a Yukawa potential in a uniform neutralizing background. This is consistent with other theoretical predictions and with observations of UCP (Gericke & Murillo Reference Gericke and Murillo2003; Chen et al. Reference Chen, Simien, Laha, Gupta, Martinez, Mickelson, Nagel and Killian2004; Foster et al. Reference Foster, Fetsch and Fisch2023).

With the potentials defined in (3.1), the electrostatic part of the electron Massieu potential, which is a function of $S_{ii}$, is

(3.4)\begin{equation} \varPhi_{\phi e}(\beta_e, V, S_{ii}) ={-} \frac{V}{2}\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} \left[ \ln \left(\frac{\kappa^2}{k^2} + 1 \right) - \frac{\kappa^2}{k^2} - \frac{1}{k^2}\frac{ Z \kappa^4}{\kappa^2 + k^2} S_{ii} \right] . \end{equation}

The electrostatic part of the ion Planck potential is

(3.5)\begin{equation} \varPsi_{\phi i}(\beta_i, V, \mathcal{F}_i) ={-} \frac{V}{2}\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} \left[\ln \left(\frac{\chi^2}{k^2} - 2\tau \mathcal{F}_e + 1 \right) - \frac{\chi^2}{k^2} \right]. \end{equation}

Finally, when the ions are at equilibrium, the electron Planck potential can be written in terms of the equilibrium ion–ion structure factor as

(3.6)\begin{align} \varPsi_{\phi e}(\beta_e,V, \mathcal{F}_e) ={-} \frac{V}{2}\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} \left[ \ln \left(\frac{\kappa^2}{k^2} + 1 \right) - \frac{\kappa^2}{k^2} - \left(\frac{1}{k^2}\frac{ Z \kappa^4}{\kappa^2 + k^2} - 2\mathcal{F}_e \right) \langle S_{ii} \rangle\right] . \end{align}

3.2. Structure factors

We can immediately substitute the above Green's function definitions, as well as the expression (3.2) that we derived for the force scaling factor, to get the structure factors from (2.40), (2.42) and (2.44). For each pair of species we have

(3.7)\begin{equation} \left.\begin{gathered} \langle S_{ii} \rangle(\boldsymbol k) = \frac{k^2 + \kappa^2}{k^2 + \kappa^2 + \chi^2} ,\\ \langle S_{ie} \rangle(\boldsymbol k) = \frac{\sqrt{Z}\kappa^2}{k^2 + \kappa^2 +\chi^2} ,\\ \langle S_{ee} \rangle(\boldsymbol k) = \frac{k^2}{k^2 + \kappa^2} + \frac{1}{\tau}\frac{\kappa^2\chi^2}{(k^2 + \kappa^2)(k^2 + \kappa^2 + \chi^2)} . \end{gathered}\right\} \end{equation}

These expressions recover the known results for structure factors (and equivalently, position-space correlation functions) in two-temperature plasma (Salpeter Reference Salpeter1963; Ecker & Kröll Reference Ecker and Kröll1964). The structure factors can be made more physically transparent when written in terms of the deviation from those of an ideal gas, where correlations between particles vanish. In a system of uncorrelated particles, the structure factor for species $a, b$ is given by $S_{ab} = \delta _{ab}$ where $\delta$ is the Kronecker delta. This comes from the fact that in the formula for the structure factor

(3.8)\begin{equation} \langle S_{ab }(\boldsymbol k) \rangle = \langle {\tilde{\rho}}^*_a {\tilde{\rho}}_{b} \rangle = \frac{1}{\sqrt{N_a N_{b}}}\sum_j \exp\{{{\rm i}\boldsymbol k \boldsymbol{\cdot} r_{a,j}}\}\sum_\ell \exp\{{-{\rm i}\boldsymbol k \boldsymbol{\cdot} r_{b,\ell}}\}, \end{equation}

the double summation includes $j=\ell$ terms, in which the same particle appears in both summations, in cases where $a = b$. Each of these terms yields 1 while the other terms, which each describe a correlation between two distinct particles, average to zero in an ideal gas. We can write the structure factors as

(3.9)\begin{equation} \left.\begin{gathered} \langle S_{ii}(\boldsymbol k) \rangle = 1 - \frac{\chi^2}{k^2 + \kappa^2 + \chi^2} ,\\ \langle S_{ie}(\boldsymbol k) \rangle = \frac{\sqrt{Z}\kappa^2}{k^2 + \kappa^2 +\chi^2} ,\\ \langle S_{ee}(\boldsymbol k) \rangle = 1 - \frac{\kappa^2}{k^2 + \kappa^2} + \frac{1}{\tau}\frac{\kappa^2\chi^2}{(k^2 + \kappa^2)(k^2 + \kappa^2 + \chi^2)} . \end{gathered}\right\} \end{equation}

For finite $k$, the deviations from ideal-gas behaviour for both electron–electron and electron–ion structure factors vanish as expected when $\kappa \rightarrow 0$, corresponding to large electron temperature. Similarly, the deviation in ion–ion structure factor vanishes as $\chi \rightarrow 0$, corresponding to large ion temperature. We note, however, that the electron–ion structure factor remains non-zero when $\chi \rightarrow 0$. This means that, although the ions become uncorrelated with each other in the large ion temperature limit, the finite-temperature electrons can still react to each ion configuration, allowing some electron–ion correlation to persist.

3.3. Energy

The ion electrostatic energy at equilibrium, calculated from (2.50), is given by

(3.10)\begin{equation} U_{\phi i} = T_i \frac{V}{4{\rm \pi}^2} \chi^2 \int_0^\infty {\rm d}k \left[\frac{-\chi^2}{k^2 + \kappa^2 + \chi^2}\right] , \end{equation}

which is easily integrated to find

(3.11)\begin{equation} U_{\phi i} ={-}T_i \frac{V\chi^4}{8{\rm \pi} k_D} . \end{equation}

For the electrons, the potential energy as a functional of $S_{ii}$ is given from (2.48) by

(3.12)\begin{equation} U_{\phi e}(T_e; S_{ii}) = T_e \frac{V}{2}\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} \frac{\kappa^2}{k^2} \left[ \frac{-\kappa^2}{k^2 + \kappa^2} + \left(\frac{Z \kappa^4}{(k^2 + \kappa^2)^2} - 2\frac{Z\kappa^4}{k^2(k^2 + \kappa^2)}\right)S_{ii}\right] . \end{equation}

For equilibrium ions, i.e. an ion structure factor given by (3.9), the electron potential energy becomes

(3.13)\begin{equation} U_{\phi e} = T_e \frac{V}{4{\rm \pi}^2} \kappa^2 \int_0^\infty {\rm d}k \left[\frac{-\kappa^2}{k^2 + \kappa^2} + Z\frac{\kappa^4}{(k^2 + \kappa^2)(k^2 + \kappa^2 + \chi^2)} - 2Z\frac{\kappa^2}{k^2 + \kappa^2 +\chi^2}\right], \end{equation}

which reduces to

(3.14)\begin{equation} U_{\phi e} = \frac{V}{8{\rm \pi}} T_e\kappa^3\left[{-}1 + Z \frac{\kappa^2 k_D - \kappa k_D^2 - \kappa \chi^2}{\chi^2 k_D}\right]. \end{equation}

Adding the energies of each species gives the total electrostatic potential energy as

(3.15)\begin{equation} U_{\phi}(T_e, T_i) = \frac{V}{8{\rm \pi}}\left[{-}T_e\kappa^3 - T_i(k_D^3 - \kappa^3)\right] . \end{equation}

This result is identical to the one obtained by Foster et al. (Reference Foster, Fetsch and Fisch2023). It also matches calculations by Triola (Reference Triola2022), discussed further below.

Although the partition of potential energy into the electron and ion subsystems given by (3.14) and (3.11) is not trivial, the combined energy has a straightforward physical interpretation. It is a well-known result (Kelly Reference Kelly1963) that in weakly coupled Coulomb-interacting plasma at thermal equilibrium, the potential energy is

(3.16)\begin{equation} U_\phi^\mathrm{equilib}(T) ={-}\frac{V}{8{\rm \pi}} T_e k_D^3 . \end{equation}

In (3.15), we can identify the term ${-(V/8{\rm \pi} )T_e\kappa ^3}$ as the potential energy of a gas of electrons interacting directly through a Coulomb potential, and identify the term ${-(V/8{\rm \pi} )T_i(k_D^3 - \kappa ^3)}$ as the potential energy of a gas of ions interacting through a shielded potential.

It is also useful to write the energy in terms of simple dimensionless parameters. As above, we write the temperature ratio as $\tau = T_e/T_i$. In terms of the plasma parameter $\varLambda = n\lambda _D^3$, where $N = (1 + Z)N_i$ and $n=N/V$, the total electrostatic energy is

(3.17)\begin{equation} U_\phi(T_e,T_i) ={-}\frac{T_i N}{8{\rm \pi}\varLambda}\left[1 + \frac{\tau - 1}{(1 + \tau Z)^{3/2}}\right] . \end{equation}

Here, the energy includes a term with the same form as the equilibrium result ${U_\phi ^\mathrm {equilib} = -TN/8{\rm \pi} \varLambda }$, plus a term proportional to $(\tau - 1)$, which clearly vanishes in the equal-temperature case.

In the regime of interest, our (3.15) is equivalent to the potential energy derived in recent work (Triola Reference Triola2022) studying equations of state in two-temperature plasma. However, an advantage of the approach here is the fact that it affords a clear prescription for partitioning the potential energy into electron and ion components. Some methods exist, but they differ depending on one's chosen scheme for assigning the ‘cross-temperature’ that applies to electron–ion interactions. Our approach, although more restricted in requiring the large mass ratio limit $m_e/m_i \ll 1$, has the advantage of providing a natural separation between electron and ion energies, given in (3.14) and (3.11). This separation is necessary when we want to predict the evolution of electron and ion temperatures in response to heating or to volume changes. In the subsequent sections, we further develop the separate thermodynamic descriptions of the two species, and then apply these descriptions to calculate changes in electron and ion temperatures.

3.4. Entropy

In order to track heat flow into the two subsystems separately, it is necessary to compute the entropy separately for each species. If both species are individually in equilibrium, we should use the generalized Gibbs ensemble and apply (2.51) to calculate both entropies. If $S_{ii}$ is known instead, then we can calculate the electron entropy in the canonical ensemble using (2.52). The calculation in either case is similar and is outlined as follows.

In the generalized Gibbs ensemble, it is first necessary to calculate explicitly the Planck potential of each species from (3.4) and (3.5). Because we will not need to take derivatives, we can substitute the force from (3.2) to obtain $\hat {\varPsi }_i(\beta _e,\beta _i,V)$ and $\hat {\varPsi }_e(\beta _e,\beta _i,V)$. For the ions first, we have

(3.18)\begin{equation} \hat{\varPsi}_{\phi i} ={-} \frac{V}{2}\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} \left[\ln(k^2 + \kappa^2 + \chi^2) - \ln(k^2 + \kappa^2) - \frac{\chi^2}{k^2}\right], \end{equation}

which evaluates to

(3.19)\begin{equation} \hat{\varPsi}_{\phi i} = \frac{V}{12{\rm \pi}}(k_D^3 - \kappa^3) . \end{equation}

The Planck potential for the electrons simplifies greatly when substituting the generalized force from (3.2), giving

(3.20)\begin{equation} \hat{\varPsi}_{\phi e} ={-} \frac{V}{2}\int \frac{{\rm d}^3k}{(2{\rm \pi})^3} \left[ \ln \left(\frac{\kappa^2}{k^2} + 1 \right) - \frac{\kappa^2}{k^2}\right] , \end{equation}

and therefore

(3.21)\begin{equation} \hat{\varPsi}_{\phi e} = \frac{V}{12{\rm \pi}}\kappa^3 . \end{equation}

For finding the entropy of each species, it is finally necessary to compute the term associated with the entropic force, which for species $s$ we will denote $\mathcal {S}_{fs}$. For the electrons, this term is defined as

(3.22)\begin{equation} \mathcal{S}_{fe} \doteq \mathcal{F}_e \cdot S_{ii} , \end{equation}

which integrates to

(3.23)\begin{equation} \mathcal{S}_{fe} = \frac{V}{8{\rm \pi}}Z\frac{\kappa^4}{k_D} . \end{equation}

The entropic force contribution $\mathcal {S}_{fi} \doteq \mathcal {F}_i \cdot S_{ii}$ to the ion entropy results from a nearly identical calculation as

(3.24)\begin{equation} \mathcal{S}_{fi} ={-}\tau \frac{V}{8{\rm \pi}}Z\frac{\kappa^4}{k_D} . \end{equation}

Then, in total, the electrostatic component $\mathcal {S}_{\phi e}$ of the electron entropy simplifies to

(3.25)\begin{equation} \mathcal{S}_{\phi e} ={-}\frac{V}{24{\rm \pi}}\kappa^3 - \frac{V}{ 8{\rm \pi}} Z \frac{\kappa^4}{k_D + \kappa}, \end{equation}

and the electrostatic component $\mathcal {S}_{\phi i}$ of the ion entropy is

(3.26)\begin{equation} \mathcal{S}_{\phi i} = \frac{V}{12{\rm \pi}}(k_D^3 - \kappa^3) - \frac{V}{8{\rm \pi} } \chi^2 k_D . \end{equation}

We can also find the electron entropy in the canonical ensemble, corresponding to a known $S_{ii}$. When the ions are at equilibrium this $S_{ii}$ is the equilibrium structure factor $\langle S_{ii} \rangle$, and so we would have $\varPhi _e(\beta _e, V, S_{ii}) = \hat {\varPsi }_e(\beta _e, V, (\hat {V} \mathcal {F}_e))$. For arbitrary ion correlations, we instead use (2.52) for the electron entropy, which along with (2.27) for the Massieu potential and (3.12) for the electron energy gives

(3.27)\begin{align} \mathcal{S}_{\phi e} & ={-}\frac{V}{2} \int \frac{{\rm d}^3k}{(2{\rm \pi})^3} \left[ \ln\left(\frac{\kappa^2}{k^2} + 1\right) - \frac{\kappa^2}{k^2} + \frac{\kappa^2}{k^2 + \kappa^2}\right.\nonumber\\ & \quad\left. - \left(\frac{Z^2\kappa^4}{k^2(k^2 + \kappa^2)} + 2\frac{Z\kappa^4}{k^2(k^2 + \kappa^2)} - \frac{Z\kappa^4}{(k^2 + \kappa^2)^2}\right) S_{ii} \right] . \end{align}

We can now use these entropy expressions to study adiabatic processes, in which the entropy of one or both species remains constant.