Classical Representation of a Quantum System at Equilibrium: Theory
A quantum system at equilibrium is represented by a corresponding classical system, chosen to reproduce thermodynamic and structural properties. The motivation is to allow application of classical strong coupling theories and molecular dynamics simulation to quantum systems at strong coupling. The correspondence is made at the level of the grand canonical ensembles for the two systems. An effective temperature, local chemical potential, and pair potential are introduced to define the corresponding classical system. These are determined formally by requiring the equivalence of the grand potentials and their functional derivatives. Practical inversions of these formal definitions are indicated via the integral equations for densities and pair correlation functions of classical liquid theory. Application to the ideal Fermi gas is demonstrated, and the weak coupling form for the pair potential is given. In a companion paper two applications are described: the thermodynamics and structure of uniform jellium over a range of temperatures and densities, and the shell structure of harmonically bound charges.
I Introduction and Motivation
The equilibrium properties (thermodynamics and structure) of classical systems for conditions of strong coupling (e.g., liquids) can be described by a number of practical and accurate approximations Hansen . Examples are those based on integral equations for the pair correlation function, such as the Percus-Yevick and hypernetted chain approximations (HNC). Thermodynamic properties are then determined from exact representations in terms of the pair correlation function. Extensions to inhomogeneous systems have been described in terms of the pair correlation function as well. More extensive formulations of the thermodynamics and structure for classical systems can be obtained within the framework of classical density functional theory Lutsko . Molecular dynamic simulation (MD) is perhaps the most accurate tool for the description of classical systems.
A corresponding practical description of strong coupling when quantum effects (diffraction and degeneracy) are important is less well-developed, except when a transformation to a corresponding weak coupling quasi-particle representation can be found (e.g., phonons, conventional Fermi liquid theory). Conditions of current interest for which such representations are not available include those of the growing class known as ”warm, dense matter”, i.e. complex ion-electron systems WDM . The ions are typically semi-classical but the electron conditions span the full range of classical plasmas to zero temperature solids. With such potential applications in mind, the objective here is to extend the classical approximation methods to include quantum effects. The idea is to introduce a representative classical statistical mechanics that embodies selected quantum effects via effective thermodynamic parameters and effective pair potentials in the classical Hamiltonian. These classical parameters and pair potential are defined formally in such a way as to ensure the equivalence of the classical and quantum thermodynamics and structure. In this way the established effective classical methods can be applied to describe quantum systems at strong coupling as well. The precise definition of the effective classical statistical mechanics is given in the next section. Of course, there is no possibility to map a quantum system entirely onto an equivalent classical system. Instead, the classical system is defined to yield only selective quantum properties, in this case the thermodynamics and pair structure. Other properties calculated from this effective statistical mechanics, such as higher order structure or transport properties, constitute uncontrolled approximations, and its utility in this more general context must be obtained from experience with applications.
The analysis here is a formalization of related partial attempts to incorporate quantum effects in classical calculations, most commonly through effective classical pair potentials. There is an extensive history for constructing such effective potentials Uhlenbeck , Morita , Kelbg , Lado , Dunn67 , Filinov , Dufty09 . Recent reviews with references can be found in reference Jones07 . Such potentials have been applied in a number of classical theories and simulations. In particular they provide a definition for the classical statistical mechanics of electron-ion systems which otherwise do not exist due to the Coulomb collapse without diffraction effects. Effective pair potentials defined from the two particle density matrix do not assure that the thermodynamics or structure will be given correctly by the corresponding classical statistical mechanics, and those approaches provide uncontrolled approximations in most applications. In contrast, the formal approach here has the advantage of predicting the correct thermodynamics and structure while retaining the simplicity of pair potentials for point particles without approximation. However, the additional complication implied by this is the need for effective classical thermodynamic parameters (e.g., temperature and chemical potential) that differ from those for the given quantum system.
The introduction of an effective temperature in this context appears to have been done first by Perrot and Dharma-wardana (PDW) in their classical map hypernetted chain model DWP00 . It provides a means to calculate a pair correlation function from the classical HNC integral equation with two modifications: 1) the actual pair potential is replaced by the sum of a pair potential to give the correct ideal gas quantum pair correlation function, plus a phenomenological modification of the actual pair interaction to account for diffraction effects (Deutsch potential Deutsch ), and 2) an effective temperature . The single new parameter is determined as a function of the density by requiring that the classical correlation energy at be the same as that for the quantum system. The latter is determined from diffusion Monte Carlo simulation data. The solution to the HNC integral equation with these embedded quantum effects combined with classical strong coupling provides an approximate from which thermodynamic properties can be calculated from their classical expressions (e.g., pressure from the virial equation). During the past decade this remarkably simple and practical approach has been applied to a number of quantum systems with surprising success for a wide range of temperatures and densities. A recent review with references is given in DW11 . These successes provide motivation and support for the work presented here, which can be considered a formalization of these concepts into a precisely defined context.
Most practical forms of present quantum many-body theories and simulation methods have limited domains of validity, for example: Fermi liquid theory (low temperature), quantum plasma methods (high temperature), RPA with local field corrections (moderate coupling), quantum density functional theory (low temperature), path integral Monte Carlo (small particle number). As noted above these limitations have become clear recently with attempts to describe new experimental conditions of ”warm, dense matter”. The classical formulation provides new access to these extreme conditions by incorporating accurately described classical strong coupling effects and hence provides a complementary new tool. In addition, applications can often be numerically simpler (e.g., classical density functional theory, molecular dynamics, classical Monte Carlo simulation).
The definition of a representative classical system in terms of selected equilibrium properties of the underlying quantum system is given in the next section. The quantum system is represented in the grand canonical ensemble. For generality an external single particle potential is included so the thermodynamics is that for an inhomogeneous system. Interactions among particles is given by a pair potential . Only a single component system is considered here for simplicity. Thermodynamic properties are functions of the temperature and functionals of the local chemical potential . A corresponding classical system is specified in a classical grand canonical ensemble in terms of an effective classical temperature , classical local chemical potential , and classical pair potential . These three classical quantities are defined by three independent constraints equating chosen classical and quantum thermodynamics and structure: equality of the grand potentials and their functional derivatives with respect to the chemical potential and pair potential for the classical and quantum systems. Equivalently, this requires that the the pressures, the local densities, and the pair correlation functions are the same for the two systems.
Having defined the classical system in terms of equating classical and quantum properties, there is then the practical problem of inverting the defining equations for the classical temperature, local chemical potential, and pair potential. Formally this is done via classical density functional theory which relates the local density and pair correlation function to the classical potential. Inverting this and exploiting the equivalence of classical and quantum pair correlation functions and densities gives the desired expression for the classical pair potential as a functional of the quantum pair correlation function and density. Examples of practical implementations of this approach are given by Percus-Yevick (PY) and HNC approximate integral equations. With the classical pair potential determined (formally), the classical temperature is obtained from the classical virial equation for the pressure and the HNC equation for the chemical potential.
In this way the classical temperature, chemical potential, and pair potential are given as explicit functionals of the quantum properties. In practice, analysis of these functionals entails the full quantum many-body problem so it would seem that little progress has been made. The key assumption in this approach is that the strong coupling effects have a dominant classical component while dominant quantum effects are local (e.g. diffraction) or global but weakly dependent on interactions (e.g. exchange degeneracy). Then, simple weak coupling approximations to the defining functionals for the classical parameters are sufficient while strong coupling effects are obtained by a more complete implementation of the classical statistical mechanics. A more ambitious approach would try to include limited strong coupling effects in the classical parameters (e.g. correlation energy as in the PDW model above). The formalism here provides a basis for approximations that can be tailored to the specific system or objective considered. A brief overview of this approach has been described elsewhere Dufty12 .
As a first application the uniform electron gas (jellium) is considered in the following companion paper Dutta12 . The effective pair potential used incorporates the exact ideal gas exchange and the random phase weak coupling limit. The pair correlation function is then obtained for this potential from the HNC integral equation describing strong classical correlations, known to be accurate for Coulomb interactions. The results are compared to earlier quantum methods Tanaka86  and the above PDW model over a range of temperatures and densities. Good agreement with diffusion MC simulations at and recent restricted path integral MC results at finite temperatures is obtained. A second application described briefly there is to charges in a harmonic trap where classical strong correlations produce shell structure. Qualitative effects of diffraction and exchange on the formation of shell structure are observed.
Ii Definition of the representative classical system
ii.1 Thermodynamics from statistical mechanics
Consider a quantum many-body system with Hamiltonian in a volume at equilibrium described by the grand canonical ensemble, with inverse temperature and chemical potential . For simplicity a one component system is considered and dependence on internal degrees of freedom such as spin is suppressed. The Hamiltonian is of the form
where denotes the total kinetic energy,
is the pair-wise additive potential energy among particles, is an external potential coupling to each particle
and is the particle number. The forms for the pair potential and single particle external potential are left general for the present (beyond well-known conditions for the existence of the grand potential defined below). The statistical density operator for the grand ensemble depends on in the form . It is convenient to combine the contributions from the external potential with that from
where is the number density operator
and is a local chemical potential
A caret over is used to distinguish the operator from its average value .
The thermodynamics of the system is determined from the grand potential , where is the grand partition function. The thermodynamic parameters for this function are the inverse temperature , the local chemical potential , and the volume . The grand potential is
The symbol denotes a trace over the particle Hilbert space with appropriate symmetry restrictions for Fermions or Bosons (equivalently, in a second quantized representation the sum over N and the represent a trace over Fock space). The vertical line in the arguments of and denotes a functional of the quantity following it, e.g. is a function of and a functional of and . Its functional dependence on the pair potential has also been made explicit in this notation, although it is not strictly a thermodynamic variable. The dependence on the volume is left implicit. The pressure is proportional to the grand potential
The first order derivatives of the grand potential give the internal energy and average number density
Higher order derivatives provide the fluctuations (susceptibilities) and structure. In particular, the second functional derivative with respect to is related to the response function
where , and denotes an equilibrium grand canonical average of the quantity . A related quantity is the pair correlation function, , obtained by functional differentiation with respect to the pair potential
For the quantum system there is no simple relationship between the two measures of structure, and . Instead, only their time dependent extensions are related via a fluctuation - dissipation relation.
A corresponding classical system is considered with Hamiltonian in the same volume at equilibrium described by the classical grand canonical ensemble, with inverse temperature and local chemical potential . The Hamiltonian has the same form as (1) except that the potential energy functions (3) are different, and denoted by
The local chemical potential is
The classical grand potential is defined in terms of these quantities by
Here, is the thermal de Broglie wavelength associated with the classical temperature. The integration for the partition function is taken over the particle configuration space.
Here denotes the corresponding classical equilibrium grand canonical average of the quantity . Note the additional constraint that the derivatives are taken at constant pair potential . This is necessary because the determination of , , and for the quantum correspondence in the next subsection implies is a function of and a functional of . This dependence is left implicit to simplify the notation. See Section below for further elaboration.
ii.2 Classical - quantum correspondence conditions
The classical system has undefined ingredients: the effective inverse temperature, , the local chemical potential, , and the pair potential for interaction among the particles, . A correspondence between the classical and quantum systems is defined by expressing these quantities as functions or functionals of , , and . This is accomplished by requiring the numerical equivalence of two independent thermodynamic properties and one structural property for the classical and quantum systems. The first two are chosen to be the equivalence of the grand potential and its first functional derivative with respect to the local chemical potential.
An equivalent form for these conditions can be given in terms of the pressure and density
These two relations provide two independent relations between of , and the physical variables and .
It remains to have a structural equivalence to relate the pair potential to , which are two particle functions. This is accomplished by equating the functional derivatives of the grand potentials with respect to these pair functions
More physically, this implies the equivalence of density fluctuations
In summary, the classical - quantum correspondence conditions are the equivalence of the pressures, densities, and pair correlation functions.
It might seem that a natural alternative choice to (25) would be to equate the second functional derivatives of the grand potential with respect to the local chemical potential, or equivalently . However, the classical response function has a singular contribution proportional to that is not present in the quantum response function. Hence their equivalence does not provide a simple mapping of the parameters. In contrast, the classical and quantum forms for the pair correlation functions are similar and do not have this problem.
ii.3 Inversion of the correspondence conditions
To determine the classical parameters, (26), inversion of the functional forms (22) and (25) is required. This is done on the basis of three exact results of classical statistical mechanics, for the pressure, density, and pair correlation function respectively. The first is the virial form for the classical pressure (obtained by differentiating with respect to the volume)
(Note that this is the average pressure over the entire system. A local pressure could be identified with the integrand of (27)). The second is an equation for the classical densities,
and the third is an equation for the pair correlation functions
The direct correlation function appearing in these equations is determined from via the Ornstein-Zernicke equations
These equations provide the inversion in the following way. First, (28) and (29) are solved for and as functionals of and . Then and are replaced by their quantum counterparts, and , according to the definitions (22) and (25), giving the desired exact expressions
Here, is defined by the Ornstein-Zernicke equation (30) with replaced by
Finally, an equation for is obtained by using the equivalence of the classical and quantum pressures , (22), to write
or with (27)
which is equivalent to (26) (corresponding to a change of variables from to for the quantum system).
These expressions for the classical parameters are exact, but deceptively simple, equations. Their complexity arises from the fact that different arguments for the functional occur. The relationship of two functionals at different arguments depends on higher order correlations that are not determined by these equations. The many-body problem has not been solved, only hidden, and approximations are required for practical applications
The application of these results proceeds in three steps. First an approximation to the classical equations (28) and (29) is chosen such that classical correlations are well described. These are then inverted to obtain the corresponding approximations to (31) and (32). Next, selected information about the quantum system is chosen in the form of approximations to and , for the calculation of the classical parameters of (36). Finally, with all classical parameters known properties of interest are determined from classical statistical mechanics (e.g., liquid state integral equations, classical density functional theory, molecular dynamics).
ii.4 Hypernetted Chain Approximation
Together with the Ornstein-Zernicke equation (30) they are a closed set of equations to determine the density and pair correlation function for given potentials. This is the HNC of liquid state theory, generalized to spatially inhomogeneous systems Attard89 . It is known to give very good results for uniform Coulomb systems Hansen  and for inhomogenous, confined Coulomb systems even at strong coupling conditions Wrighton09 , Wrighton12 .
and the Ornstein-Zernicke equation (32a) is unchanged
The ”hole function”, has been introduced for notational simplicity
These equations, together with (35) for , provide practical forms to determine , , and for the classical system, given appropriate quantum input.
In the uniform limit (no external potential) the second term of (40) is simply related to the static structure factor
For Coulomb systems, such as jellium considered in the following companion paper Dutta12  and ideal Fermi fluids at zero temperature, vanishes as and this contribution to diverges. Therefore, instead of (40) an alternative form obtained from a coupling constant integration and the HNC Baus  is used,
Iii Thermodynamics - further considerations
The definition of the equivalent classical system assures that the pressure and density in the grand ensemble give the correct quantum results (in principle), e.g.
However, some care is required in the definition of other thermodynamic properties. For the quantum system the thermodynamic variables are for constant . However, their variation leads to a change in all three classical variables , since all are functions of and functionals of . Therefore, the variation of the pressure leads to (in the following the volume is always held constant)
The second equality defines the classical thermodynamic entropy and thermodynamic density in terms of the grand potential
Similarly, the classical internal energy is defined by
These same relationships hold for the quantum properties as well, since they are the general definitions of thermodynamics for the chosen variables, . In the quantum case is independent of the thermodynamic variables and hence is constant in the variations. This leads to the equivalent expressions in terms of equilibrium averages
However, in the classical case the above leads to
For example, the thermodynamic density, differs from the average density of the text above, , because the latter is defined as a derivative of the grand potential at constant .
Iv Example - Inhomogeneous Ideal Fermi Gas
To illustrate this definition of an equivalent classical statistical mechanics, consider the case of non-interacting Fermions in an external potential
Since is now the sum of single particle operators, the pressure, density, and pair correlation function can be expressed in the form of single particle calculations
where denotes a matrix element in coordinate representation, and is the spin.
Further reduction of these results to expose the dependence on and is difficult without solving the eigenvalue problem for the single particle Hamiltonian . A useful practical approximation that captures most of the important exchange effects (but not any bound states if supported by the external potential) is obtained by replacing the operator by its eigenvalue , a ”local density approximation”. The expectation values above then can be calculated as simple integrals
The results (58) are the familiar finite temperature Thomas-Fermi approximations for the thermodynamics, while (60) is its extension to structure DT11 . The expressions for and are no longer functionals of , but rather local functions of and , respectively. The change of variables from to is accomplished by inverting (59) to obtain . Practical fits for and this inversion are recalled in Appendix B, along with simplification of .
The corresponding classical results are non-trivial because when , because classical pair interactions are required to reproduce quantum exchange effects. Thus the thermodynamics and structure of a simple ideal quantum gas requires a corresponding classical system with the full complexity of an interacting many-body system. This classical many-body problem is addressed here and below in the HNC approximation described above by (40) and (41) together with the Ornstein - Zernicke equation (42) and (35) for . The solution for the classical parameters is a straight forward numerical task, but the results are different for each given external potential. To simplify the illustration here the following is further restricted to a uniform ideal Fermi gas () in the thermodynamic limit. See section 5 below for the non-uniform case of charges in a harmonic trap. For the uniform gas the equations for the classical system parameters simplify to
The superscript on and the dependence of on thermodynamic variables has been suppressed for simplicity. These last two equations can be solved for using from (57) and (60) in the uniform limit. With that result, can be calculated from (61), and then determined from (45). Further elaboration is given in Appendix B.
The dimensionless potential will be referred to as the Pauli potential. It is shown in Figure 1 as a function of the dimensionless coordinate where is the mean distance between particles defined by . The state conditions are represented by characterizing the density (where is the Bohr radius), and for the temperature relative to the Fermi temperature (). For example, in these units and the classical limit occurs for where the distance between particles is large compared to the thermal de Broglie wavelength. It is noted that ideal gas properties expressed in terms of and become independent of Figure 1 shows the Pauli potential at for , , , and .
Generally, the potential is positive, finite at (Pauli exclusion), and monotonically decreasing. The behavior is exponential at small , but an algebraic tail develops for small . This arises from the direct correlation function in (63). Classical statistical mechanics does not exist for such a long range potential and it would appear that the equivalent classical system proposed here fails even for this simplest case of an ideal Fermi gas. However, this problem is ”cured” for the corresponding case of classical Coulomb interactions with the same long range problem by adding a uniform neutralizing background (the one component plasma). The same procedure can be used here, i.e., a classical system is considered where the pair potential is supplemented in the Hamiltonian by a corresponding uniform compensating background. The pressure equation (27) is modified due to this background by the replacement of by , and (61) becomes
Figure 2 shows the classical temperature relative to the Fermi temperature, , as a function of obtained from (65). It is seen that the classical temperature remains finite at in all cases, and crosses over to at high temperatures. The PDW model postulates the form . The model originally uses the average energy per particle at to evaluate . The result from (65) is quite close To compare the dependence at finite , the PDW form is also shown in Figure 2 with . It is seen that the results are quite similar although the PDW form has a somewhat faster cross over to the classical limit.