In general, we distinguish two types of phase transitions: first-order and second-order (Ehrenfest classification), first-order transitions are discontinuous in the first derivative of free energy F, second-order are continuous in the first derivative and discontinuous in the second derivative ( [1] chap. 1.1).

The derivatives are in an order variable of the system, in solid-fluid-gas PT’s it is usually volume V or density $\rho =\frac{N}{V}$, in magnetic PT’s it is the magnetization M.

These apply mostly to the solid-fluid-gas transitions, where the discontinuous

variable is the volume $V=\frac{\partial G}{\partial p}$, or equivalently the average distance λ.

Here the average distance λ, the density ρ, specific volume $v=\frac{V}{N}$ and heat capacity C_v are discontinuous.

Transitions happen at the melting temperature T_m, and at the boiling temperature T_b, there is a latent heat $\Delta C$ at the transition temperature.

These apply mostly to the phase transitions in magnetic materials (ferromagnetic, antiferromagnetic) and in superconducting materials.

In magnetic materials, the magnetization $M={\left(\frac{\partial F}{\partial B}\right)}_T$ is continuous at Curie temperature T_C: in ferromagnetic phase transition at T_C spontaneous magnetization $M={\left(\frac{\partial F}{\partial B}\right)}_T$ becomes 0, and there is no latent heat.

The magnetic susceptibility ${\chi }_T={\left(\frac{{\partial }^{2}F}{\partial B^2}\right)}_T$ is discontinuous at T_C.

In superconducting materials, the density n_s of superconducting electrons is continuous at critical temperature T_c, where it becomes zero. The coherence length at small perturbations $\xi$ is the analogue of the susceptibility: it is discontinuous at the critical temperature T_C.

Maxwell’s equal-area rule in first-order transitions

When $F\left(\lambda ,\beta \right)$ has a maximum in λ, then pressure

$p\left(\lambda ,\beta \right)=-\frac{1}{4\pi {\lambda }^2}\frac{\partial F\left(\lambda ,\beta \right)}{\partial \lambda }$ has negative-zero-positive transition, and a region of negative pressure is unstable, therefore there is a “jump” in λ over this area, i.e. a phase transition. The transition is determined by Maxwell’s equal-area rule $p\left(v_1,T\right)=p\left(v_2,T\right)$ and ${\displaystyle \underset{v_1}{\overset{v_2}{\int }}p\left(v,T\right)dv}=0$ (equivalently $F\left(v_1,T\right)=F\left(v_2,T\right)$)

So we have the following behavior for phase transitions:

fluid-->gas at saturation curve:

${\lambda }_1={\lambda }_f\left(T\right)\to {\lambda }_2={\lambda }_g\left(T\right)$ determined by Maxwell’s equal-area rule

$F\left(v_1,T\right)=F\left(v_2,T\right)$, $p\left(v_1,T\right)=p\left(v_2,T\right)$ where $p\left(v,T\right)=-\frac{\partial F\left(v,T\right)}{\partial v}$

fluid-->solid at fluid-solid line,

${\lambda }_1={\lambda }_f\left(T\right)\to {\lambda }_2={\lambda }_s\left(T\right)$ determined by Maxwell’s equal-area rule

$F\left(v_1,T\right)=F\left(v_2,T\right)$, $p\left(v_1,T\right)=p\left(v_2,T\right)$ where $p\left(v,T\right)=-\frac{\partial F\left(v,T\right)}{\partial v}$.

Heat capacity behavior in continuous magnetic phase transitions

Heat capacity C_v often diverges in the neighborhood of T_c as

$C~\left|T-T_c\right|{}^{-\alpha }$, with $0<\alpha <1$ mostly $\alpha \ll 1$

sometimes $C~\log \left(\frac{T_c}{\left|T-T_c\right|}\right)$

in general $C\left(T\right)~\frac{1}{\alpha }\left({\left|\frac{T-T_c}{T_c}\right|}^{-\alpha }-1\right)$, C finite, dC/dT need not be finite.

$U\left(r\right)=\frac{k{q}_1{q}_2}{r}$ (1)

k= 2.31 × 10⁻²⁸ J·m, q_i in e₀-units.

For screened Coulomb potential

$U\left(r\right)=A\exp \left(-r/l_D\right)/r$ (2)

where l_D=Debye length.

London potential between spheres with radii R₁, R₂

$U\left(z;R_1,R_2\right)=-\frac{A}{6}\left(\frac{2R_1{R}_2}{z^2-{\left(R_1+R_2\right)}^2}+\frac{2R_1{R}_2}{z^2-{\left(R_1-R_2\right)}^2}+\log \left(\frac{z^2-{\left(R_1+R_2\right)}^2}{z^2-{\left(R_1-R_2\right)}^2}\right)\right)$ (3)

$z=R_1+R_2+r$, $A={10}^{-20}{\mathrm{...10}}^{-19}\text{J}$

Lennard-Jones potential

$U_{LJ}\left(r,\sigma ,\epsilon \right)=4\epsilon \left({\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6\right)$ (4)

for argon ${\sigma }_{LJ}\left(Ar\right)=3.4\stackrel{\dot{}}{\text{A}}$, $\epsilon \left(Ar\right)=0.0123\text{eV}$, $p_0\left(Ar\right)=\frac{\epsilon }{{\sigma }^3}=689\text{MPa}$

Morse potential

$U_M\left(r,\sigma ,\epsilon \right)=\epsilon \left({\left(1-\exp \left(-\frac{r-\sigma }{\sigma }\right)\right)}^2-1\right)$ (5)

For arbitrarily positioned dipoles, with angle θ₁ resp. θ₂ to center connection line,

$U\left(r\right)=-\frac{e^2{r}_1{r}_2{q}_1{q}_2}{r^3}\left(-\cos \left({\theta }_1-{\theta }_2\right)+3\cos \left({\theta }_1\right)\cos \left({\theta }_2\right)\right)$ (6)

when both dipoles perpendicular to connection line, inverse direction:

${\theta }_1=\pi /2$, ${\theta }_2=-\pi /2$, $V\left(r\right)=-\frac{e^2{r}_1{r}_2{q}_1{q}_2}{r^3}$

with thermal screening

$U\left(r\right)=-\frac{2{\left(e^2{r}_1{r}_2{q}_1{q}_2\right)}^2}{3r^6}\frac{1}{k_{B}T}$ in cgs.

These are the typical first-order phase transitions found in nature ( [2] - [5] ).

The corresponding pressure-temperature phase diagram p(T) consists of three regions: solid, liquid, gas ( Figure 1 ). The saturation curve between liquid and gas region starts with the branching triple point, and ends at the critical point, where the separation of fluid and gas ceases to exist.

The corresponding pressure-volume (isotherm) phase diagram p(T) describes the two-phase transition liquid-gas along the saturation curve ( Figure 2 ).

In the two-phase region the volume “jumps” from the liquid to the gas phase at constant pressure. The isotherm part between the minimum and the following maximum is forbidden, since the compressibility there is negative.

The transition pressure is defined by the Maxwell-area-rule, as depicted schematically below.

In the Maxwell-area-rule, the area above and below the pressure is equal, i.e. $p\left(v_1,T\right)=p\left(v_2,T\right)$, $F\left(v_1,T\right)=F\left(v_2,T\right)$: pressure and free energy at transition points are equal.

At the triple point [6] , the saturation curve ends, fluid-->gas and fluid-->solid curves meet,

i.e. $p\left(v_g,T\right)=p\left(v_f,T\right)=p\left(v_s,T\right)$

The triple point can be found as the branching point of saturation curve in the $p\left(\lambda ,\beta \right)$ diagram.

At the critical point [6] , only one phase exists, the isotherm has a turning point there.

$\frac{\partial p}{\partial v}=0$, $\frac{{\partial }^{2}p}{\partial v^2}=0$

At liquid-gas critical point:

compressibility diverges ${\kappa }_T={-\frac{1}{V}\frac{\partial V}{\partial p}|}_T~{\left|T-T_c\right|}^{-\gamma }$

specific volume jump goes to zero $v=\frac{V}{N}$, $v_g-v_l~{\left|T-T_c\right|}^{\beta }$

Heat capacity diverges $C_v={-T\frac{{\partial }^{2}F}{\partial T^2}|}_V~{\left|T-T_c\right|}^{-\alpha }$

pressure derivative is discontinuous $\left|p-p_c\right|~{\left|V-V_c\right|}^{\delta }$

The critical exponents $\alpha ,\beta ,\gamma ,\delta$ obey the interdependence laws like in the magnetic phase transitions.

Critical exponents describe the thermal behavior at characteristic points in form of power law with universal (=substance-independent) exponent.

The power law has the summary form $\left|\Delta f_d\right|~{\left|\Delta f_f\right|}^{\epsilon },T\to T_c$, where $f_f$ is a fundamental variable temperature T (resp. $\beta$) or volume V (resp. $\lambda$), and $f_d$ is a dependent variable like pressure p, density ρ, compressibility κ_T, heat capacity C_V, considered at a characteristic point with temperature $T_c$.

In case of second-ordered derivatives like heat capacity $C_v\equiv {\left(\frac{\partial U}{\partial T}\right)}_V=k_B{\beta }^2{\left(\frac{{\partial }^2\log Z}{\partial {\beta }^2}\right)}_V$, the power law can be divergent i.e. ε < 0.

The universal character of critical exponents results from the fact, that the partition function can always be formulated in substance-independent and dimensionless way, using as units the fundamental energy ${\epsilon }_0$ and the molecular (hard-core) radius ${\sigma }_0$ from the intermolecular potential. Examples are the Lennard-Jones fluid and the vdWaals fluid.

We distinguish 6 critical exponents in fluid/magnetic systems, which characterize the behavior of thermodynamic variables near critical temperature T_c, correlation length ξ, volume V resp. magnetic field flux B, pressure p resp. magnetization M.

They are defined as follows in Table 1 ( [1] chap.5).

Values of 6 critical exponents are shown in the following Table 2 .

Correlation function or radial distribution function (rdf) g(r) is a measure of the probability that a particle will be located a distance r from another particle ( [7] - [9] ), it obeys the normalization condition

$4\pi {\displaystyle \underset{0}{\overset{\infty }{\int }}dr}{r}^{2}g\left(r\right)=N-1\approx N$, ${\displaystyle \underset{0}{\overset{\infty }{\int }}dr}{r}^{2}g\left(r\right)=\frac{V}{4\pi }$ (7)

Given a potential $U\left({\stackrel{\to }{r}}_1,\cdots ,{\stackrel{\to }{r}}_N\right)$, we obtain the Hamiltonian

$H={\displaystyle \underset{i=1}{\overset{3N}{\sum }}\frac{p_i^2}{2m}}+U\left({\stackrel{\to }{r}}_1,\cdots ,{\stackrel{\to }{r}}_N\right)$ (8)

The corresponding partition function reads

$\begin{array}{c}Z\left(N,V,T\right)=\frac{1}{N!h^{3N}}{\displaystyle \int d^{3N}p{d}^{3N}r\exp \left(-\beta {\displaystyle \underset{i=1}{\overset{3N}{\sum }}\frac{p_i^2}{2m}}\right)\exp \left(-\beta U\left({\stackrel{\to }{r}}_1,\cdots ,{\stackrel{\to }{r}}_N\right)\right)}\\ =\frac{1}{N!h^{3N}}{\displaystyle \int d{r}_1\cdots d{r}_N\exp \left(-\beta U\left({\stackrel{\to }{r}}_1,\cdots ,{\stackrel{\to }{r}}_N\right)\right)}=\frac{Z_N}{N!h^{3N}}\end{array}$ (9)

with thermal wavelength $\zeta =\sqrt{2\pi }\hslash c\sqrt{\frac{\beta }{m{c}^2}}=\sqrt{\frac{\beta h^2}{2\pi m}}$,

example:

$m\left({\text{O}}_{\text{2}}\right)=16\text{GeV}$, $\beta \left(T=300\text{K}\right)=1/0.026\text{eV}$, $\zeta =0.14\stackrel{\dot{}}{\text{A}}$

For the ideal gas $U=0$

$Z\left(N,V,T\right)=\frac{1}{N!}{\left(\frac{V}{h^3}{\left(\frac{2\pi m}{\beta }\right)}^{3/2}\right)}^N=\frac{1}{N!}{\left(\frac{V}{{\zeta }^3}\right)}^N$ (10)

The pressure becomes $p=\frac{1}{\beta }\frac{\partial \log Z}{\partial V}$,

$\begin{array}{c}\log Z=\left(N\left(\log V-3\log \zeta \right)-\log N!\right)\\ \approx \left(N\left(\log V-3\log \zeta \right)-N\left(\log N-1\right)\right)\end{array}$

$p\beta =\frac{\partial \log Z}{\partial V}=\frac{\partial }{\partial V}\left(N\left(\log V-3\log \zeta \right)-\log N!\right)$

because of the Stirling formula $N!\approx \sqrt{2\pi N}{\left(\frac{N}{e}\right)}^N$

$\log N!=N\left(\log N-1\right)+\frac{1}{2}\log N+\frac{1}{2}\log \left(2\pi \right)\approx N\left(\log N-1\right)$,

$p\beta =\frac{\partial \log Z}{\partial V}=\frac{N}{V}=\frac{1}{v}=\rho$, or $p\beta v=1$, $p\beta =\rho$ (11)

where $v=\frac{V}{N}={\lambda }^3=\frac{1}{\rho }$ is the specific volume, and λ is the average distance.

We obtain for the energy of ideal gas

$E=-\frac{\partial \log Z}{\partial \beta }=\frac{3}{2}\frac{N}{\beta }$ (12b)

so the specific energy (= per particle) becomes

${\rho }_E=\frac{E}{N}=\frac{3}{2\beta }$ (12a)

Average energy with potential

With the configurational partition function

$Z_N={\displaystyle \int d^3{r}_1\cdots d^3{r}_N\exp \left(-\beta U\left({\stackrel{\to }{r}}_1,\cdots ,{\stackrel{\to }{r}}_N\right)\right)}$ (13a)

we obtain

$\log Z=\log Z_N-3\log \zeta -\log N!$ (13b)

So the energy becomes

$\begin{array}{c}E=-\frac{\partial \log Z}{\partial \beta }\\ =\frac{3N}{\zeta }\frac{\partial \zeta }{\partial \beta }-\frac{1}{Z_N}{\displaystyle \int d{r}_1\cdots d{r}_{N}U\left({\stackrel{\to }{r}}_1,\cdots ,{\stackrel{\to }{r}}_N\right)\exp \left(-\beta U\left({\stackrel{\to }{r}}_1,\cdots ,{\stackrel{\to }{r}}_N\right)\right)}\\ =\frac{3}{2}\frac{N}{\beta }+\langle U\rangle \end{array}$ (14)

where $\langle U\rangle =\frac{N^2}{2V}{\displaystyle \underset{0}{\overset{\infty }{\int }}dr4\pi r^{2}u\left(r\right)g\left(r\right)}$ (grand canonical ensemble)

$E=\frac{3}{2}\frac{N}{\beta }+2\pi \frac{N^2}{V}{\displaystyle {\int }_0^{\infty }dr{r}^{2}u\left(r\right)g\left(r\right)}$

so ${\rho }_E=\frac{E}{N}=\frac{3}{2\beta }+2\pi \rho {\displaystyle {\int }_0^{\infty }dr{r}^{2}u\left(r\right)}g\left(r\right)$, $\frac{E}{\epsilon N}=\frac{3}{2}\frac{1}{\beta \epsilon }+2\pi \rho {\displaystyle {\int }_0^{\infty }dr{r}^2{u}_0\left(r\right)g\left(r,\beta \epsilon ,\rho \right)}$ (15)

where ε is the characteristic energy, and the dimensionless potential is

$u_0\left(r\right)=\frac{u\left(r\right)}{\epsilon }$

Partition function Z in terms of g(r)

$\frac{\log Z\left(N,V,\beta \right)}{N}=-\frac{1}{N}{\displaystyle \int d{\beta }_0\frac{E}{\epsilon }}=-{\displaystyle \int d{\beta }_0\left(3/\left(2{\beta }_0\right)+2\pi \frac{1}{V}{\displaystyle {\int }_0^{\infty }dr{r}^2{u}_0\left(r\right)g\left(r,\rho ,{\beta }_0\right)}\right)}$ (16)

with the dimensionless variable ${\beta }_0=\beta \epsilon$, reformulated it becomes

$\frac{\log Z\left(\rho ,{\beta }_0\right)}{N}=-\frac{3}{2}\log {\beta }_0-2\pi \frac{1}{V}{\displaystyle \int d{\beta }_0{\displaystyle {\int }_0^{\infty }dr{r}^2{u}_0\left(r\right)g\left(r,\rho ,{\beta }_0\right)}}$.

In general, the temperature dependence of the rdf is weak

$g\left(r,\rho ,{\beta }_0\right)\approx g\left(r,\rho \right)$ and the average density is constant ${\rho }_0=\frac{1}{v}=\frac{N}{V}$, so we

obtain with

$\frac{\log Z\left(\rho ,\beta \epsilon \right)}{N}\approx -\frac{3}{2}\log \beta \epsilon -\frac{\beta \epsilon }{2V}{\displaystyle \int dV{u}_0\left(r\right)g\left(r\right){\rho }_0}=-\frac{3}{2}\log \beta \epsilon -\frac{\beta \epsilon }{2}{\displaystyle \int dv{u}_0\left(r\right)g\left(r\right)}$

The pressure becomes $p=\frac{1}{\beta }\frac{\partial \log Z}{\partial V}$ ( [8] chap. 9)

$\frac{1}{Z_N}\frac{\partial \log Z_N}{\partial V}=\frac{N}{V}+\frac{\beta }{3V}\langle {\displaystyle \underset{i}{\sum }{\stackrel{\to }{r}}_i\cdot {\stackrel{\to }{F}}_i}\rangle$

with pair potential force ${\stackrel{\to }{F}}_{12}=-\frac{\partial U}{\partial {\stackrel{\to }{r}}_{12}}=-u^{\prime }\left(\left|{\stackrel{\to }{r}}_1-{\stackrel{\to }{r}}_2\right|\right)\left(\frac{{\stackrel{\to }{r}}_1-{\stackrel{\to }{r}}_2}{\left|{\stackrel{\to }{r}}_1-{\stackrel{\to }{r}}_2\right|}\right)=-u^{\prime }\left(r_{12}\right)\frac{{\stackrel{\to }{r}}_{12}}{r_{12}}$ follows

$\frac{\beta }{3V}\langle {\displaystyle \underset{i}{\sum }{\stackrel{\to }{r}}_i\cdot {\stackrel{\to }{F}}_i}\rangle =\frac{\beta N^2}{6V^2}{\displaystyle \underset{0}{\overset{\infty }{\int }}dr4\pi r^3{u}^{\prime }\left(r\right)g\left(r\right)}$

The pressure becomes

$\frac{p}{kT}=\rho -\frac{{\rho }^2}{6kT}{\displaystyle \underset{0}{\overset{\infty }{\int }}dr4\pi r^3{u}^{\prime }\left(r\right)g\left(r;kT\right)}$, $g=g\left(r,\rho ,kT\right)$, $g\left(r;\rho ,kT\right)={\displaystyle \underset{j=0}{\overset{\infty }{\sum }}{\rho }^j{g}_j\left(r;kT\right)}$

reformulated

$p\beta =\rho -\frac{{\rho }^2\beta \epsilon }{6}{\displaystyle \underset{0}{\overset{\infty }{\int }}dr4\pi r^3{u^{\prime }}_0\left(r\right)g\left(r,\beta \epsilon ,\rho \right)}$ (17)

Compare: vdWaals eos reads $\frac{p}{kT}=\frac{\rho }{1-b\rho }-\frac{a{\rho }^2}{kT}$, $v=\frac{V}{N}=\frac{1}{\rho }$, $b=2\pi {\sigma }^3/3$, resp. $p\beta =\frac{\rho }{1-b\rho }-a{\rho }^2\beta$,

from which follows for a-parameter $a\left(\rho ,kT\right)=\frac{1}{6}{\displaystyle \underset{0}{\overset{\infty }{\int }}dr4\pi r^3{u}^{\prime }\left(r\right)g\left(r,\rho ,kT\right)}$

$p\beta =\rho +{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}{B}_{j+2}\left(T\right){\rho }^{j+2}}$, with $B_{j+2}\left(T\right)=-\frac{1}{6kT}{\displaystyle \underset{0}{\overset{\infty }{\int }}dr4\pi r^3{u}^{\prime }\left(r\right)g_j\left(r;kT\right)}$

The Ornstein-Zernike equation calculates the direct correlation function c(r), which describes the pure correlation of a molecule with a neighboring molecule at distance ${\stackrel{\to }{r}}_{12}$, whereas the total correlation function g(r) (= radial distribution function) takes into account also the correlation of the neighboring molecule with a third molecule at ${\stackrel{\to }{r}}_{23}$, and all following molecules in the correlation chain.

The Ornstein-Zernike equation is an integral equation of the form [10]

$h\left(r_{12}\right)=c\left(r_{12}\right)+\rho {\displaystyle \int d^3{r}_3}c\left(r_{13}\right)h\left(r_{32}\right)$, where $h\left(r_{12}\right)=g\left(r_{12}\right)-1$ (18)

with direct correlation function c(r) and total correlation function (=radial distribution function) g(r) and with Fourier-transforms

$\hat{h}\left(\stackrel{\to }{k}\right)=\frac{\hat{c}\left(\stackrel{\to }{k}\right)}{1-\rho \hat{c}\left(\stackrel{\to }{k}\right)}$, $\hat{c}\left(\stackrel{\to }{k}\right)=\frac{\hat{h}\left(\stackrel{\to }{k}\right)}{1+\rho \hat{h}\left(\stackrel{\to }{k}\right)}$

Hypernetted-chain equation

Hypernetted-chain equation (HC equation) is a closure relation to solve the Ornstein-Zernike equation (HCOZ equation) which relates the direct correlation function to the total correlation function.

$\log g\left({\stackrel{\to }{r}}_{12}\right)+\beta u\left({\stackrel{\to }{r}}_{12}\right)=\rho {\displaystyle \int \left(h\left({\stackrel{\to }{r}}_{13}\right)-\log g\left({\stackrel{\to }{r}}_{13}\right)-\beta u\left({\stackrel{\to }{r}}_{13}\right)\right)h\left({\stackrel{\to }{r}}_{23}\right)}{d}^3{\stackrel{\to }{r}}_{23}$ (19a)

where ρ = N/V is the number density of molecules, $h\left(r\right)=g\left(r\right)-1$, g(r) is the radial distribution function, u(r) is the direct interaction potential between pairs, β=1/k_BT, and under the integral ${\stackrel{\to }{r}}_{13}={\stackrel{\to }{r}}_{12}+{\stackrel{\to }{r}}_{23}$.

HC equation yields the correlation function $g\left(r,\beta \right)$, from the interaction potential u(r).

with $r_0=\left|{\stackrel{\to }{r}}_{12}\right|$, $r_1=\left|{\stackrel{\to }{r}}_{23}\right|$, $R=\left|{\stackrel{\to }{r}}_{13}\right|$, and from cosine theorem $R_1^2=r_0^2+r_1^2{\cos }^2\theta +2r_0{r}_1\cos \theta \cos \varphi$, $R^2=r_1^2{\sin }^2\theta +R_1^2=r_0^2+r_1^2+2r_0{r}_1\cos \theta \cos \varphi$ ( Figure 3 ).

We reformulate HC equation

$\begin{array}{l}\log g\left(r_0\right)+\beta u\left(r_0\right)\\ =\rho \left(1-g\left(r_0\right)\right){\displaystyle \int \left(\left(1-g\left(R\right)\right)-\log g\left(R\right)-\beta u\left(R\right)\right)}{r}_1{}^{2}d{r}_1\sin \theta d\theta d\varphi \end{array}$

concisely $\log g\left(r_0\right)+\beta u\left(r_0\right)-\rho \left(1-g\left(r_0\right)\right)I\left(g,\beta u\right)=0$ (19b)

$I\left(g,\beta u\right)={\displaystyle \int \left(\left(1-g\left(R\right)\right)-\log g\left(R\right)-\beta u\left(R\right)\right)}{r}_1^{2}d{r}_1\sin \theta d\theta d\varphi$

where $\rho =\frac{N}{V}$, for ideal gas ${\rho }_0=\rho \left(T=273\text{K},p=1\text{at}\right)=2.69\times {10}^{25}{\text{m}}^{-3}$, or in

angstrom $\stackrel{\dot{}}{\text{A}}={10}^{-10}\text{m}$, ${\rho }_0=2.69\times {10}^{-5}{}^{\stackrel{\dot{}}{\text{A}}}\ll 1$ and mean distance ${\lambda }_0=\lambda \left(T=273\text{K},p=1\text{at}\right)=33\stackrel{\dot{}}{\text{A}}$ in the limit of small $\rho \ll 1$ $g\left(r_0\right)=\exp \left(-\beta u\left(r_0\right)\right)\approx 1-\beta u\left(r_0\right)$.

Lennard-Jones potential reads [11]

$u_{LJ}\left(r,\sigma ,\epsilon \right)=4\epsilon \left({\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6\right)$

σ is the van der Waals radius = distance at which u = 0, $\sigma =r_0$, where r₀ = half molecule diameter, r is the distance between particles.

The correlation function (=radial distribution function rdf) is here a decaying harmonics ( Figure 4 ), with a sharp first maximum at $r=\sigma$, and an amplitude of ~4x the second maximum amplitude.

The virial equation ( [8] [12] chap. 1.2) is a generalization of the vdWaals equation, where the ideal gas law is expanded in a power series in specific volume v, or in

particle density $\rho =\frac{1}{v}$.

The ideal gas law $p\beta v=\frac{p\beta }{\rho }=1$ can be formulated as an approximate invariant $E_c\left(v,\beta \right)=p\beta v$. Now we expand E_c in a power series in specific volume v, or in particle density $\rho =\frac{1}{v}$.

$E_c\left(v,\beta \right)=1+\frac{B_1\left(\beta \right)}{v}+\frac{B_2\left(\beta \right)}{v^2}+\cdots$

resp. $E_c\left(\rho ,\beta \right)=1+B_1\left(\beta \right)\rho +B_2\left(\beta \right){\rho }^2+\cdots$

The virial equation in integral reads

$p\beta =\rho \left(1-\frac{\rho \beta }{6}{\displaystyle \int d^{3}rg\left(\stackrel{\to }{r}\right)\stackrel{\to }{r}\cdot \nabla u\left(\stackrel{\to }{r}\right)}\right)$ virial equation (20a)

In comparison with vdWaals equation $p\beta =\frac{\rho }{1-\rho b}-a{\rho }^2\beta$ we see that it is

the virial expansion truncated at first order, and with a volume correction.

We can reformulate the virial equation

$p\beta =\frac{\rho }{1-\rho b}-\frac{{\rho }^2\beta }{6}{\displaystyle \int d^{3}rg\left(\stackrel{\to }{r}\right)\stackrel{\to }{r}\cdot \nabla u\left(\stackrel{\to }{r}\right)}$ virial with vdWaals-correction $b=2\pi {\sigma }^3/3$ (20b)

$p\beta =\frac{\rho }{1-\rho \left(2\pi {\sigma }^3/3\right)}-\frac{{\rho }^2\beta }{6}{\displaystyle \underset{\sigma }{\overset{\infty }{\int }}dr4\pi r^3{u}^{\prime }\left(r\right)g\left(r\right)}$ spherical symmetry, with vdWaals-correction (20c)

The vdWaals equation can be derived from the hard-sphere potential

$u_0\left(r\right)=\{\begin{array}{l}0r>\sigma \\ \infty r\le \sigma \end{array}$, $g_0\left(r\right)=\{\begin{array}{l}1r>\sigma \\ 0r\le \sigma \end{array}$, with excluded volume per particle $b=\frac{2}{3}\pi {\sigma }^3$.

The partition function is

$Z\left(N,V,T\right)=\frac{1}{N!h^{3N}}{\displaystyle \int d{r}_1\cdots d{r}_N\exp \left(-\beta U\left({\stackrel{\to }{r}}_1,\cdots ,{\stackrel{\to }{r}}_N\right)\right)}=\frac{Z_N}{N!h^{3N}}$

with 0-order including own-volume correction $Z_N^{\left(0\right)}={\left(V-Nb\right)}^N$,

$\begin{array}{c}\log Z=\left(N\left(\log V-3\log \varsigma \right)-\log N!\right)\\ \approx \left(N\left(\log V-3\log \varsigma \right)-N\left(\log N-1\right)\right)\end{array}$

The free energy becomes $F\left(N,V,T\right)\approx -\frac{1}{\beta }\log \left(\frac{{\left(V-Nb\right)}^N}{N!{\varsigma }^{3N}}\right)-a\frac{N^2}{V}$

and specific free energy (per particle) with specific volume v and density ρ

$\begin{array}{c}f\left(v,T\right)=\frac{F\left(N,v,T\right)}{N}\approx -\frac{1}{N\beta }\log \left(\frac{N{\left(v-b\right)}^N}{N!{\varsigma }^{3N}}\right)-\frac{a}{v}\\ =-\frac{1}{\beta }\left(\log \left(\frac{v-b}{{\varsigma }^3}\right)-\left(\log N-1\right)\right)-\frac{a}{v}\end{array}$

The pressure results as

$p=-\frac{\partial F}{\partial V}=-\frac{\partial f}{\partial v}=\frac{1}{\beta \left(v-b\right)}-\frac{a\beta }{v^2}=\frac{\rho }{\beta \left(1-\rho b\right)}-a\beta {\rho }^2$ (21)

which is the vdWaals equation.

The modified virial eq. reads $p\beta =\frac{\rho }{1-\rho b}-\frac{{\rho }^2\beta }{6}{\displaystyle \underset{\sigma }{\overset{\infty }{\int }}dr4\pi r^3{u}^{\prime }\left(r\right)g\left(r\right)}$, vdWaals eq. is $p\beta =\frac{\rho }{1-\rho b}-a{\rho }^2\beta$ thus ${\displaystyle \underset{\sigma }{\overset{\infty }{\int }}dr{r}^3{u}^{\prime }\left(r\right)g\left(r\right)}=\frac{3a}{2\pi }$, ${\lim }_{r\to \infty }g\left(r\right)=1$, ${\lim }_{r\to \sigma +}g\left(r\right)=0$

A possible ansatz for g(r) is

$g\left(r\right)={\Theta }_H\left(r-\sigma \right){\displaystyle \underset{k}{\sum }\left({\alpha }_k\cos \left(\frac{2k}{\pi }\frac{r-\sigma }{\sigma }\right)\right)}$ (22)

or explicitly adapted for Lennard-Jones potential

$\begin{array}{l}{g}_{LJ}\left(r,\sigma ,dr,{\alpha }_1,k,ex\right)\\ =\left(1-\frac{1}{1+\exp \left(\frac{r-\sigma }{dr\sigma }\right)}\right)\left(1+\exp \left(-\frac{exr}{\sigma }\right)\left(1-\frac{1}{1+\exp \left(\frac{r-\sigma }{dr\sigma }\right)}\right)\left({\alpha }_1\cos \left(\frac{2k}{\pi }\frac{r-\sigma }{\sigma }\right)\right)\right)\end{array}$

Where step-up-function ${\Theta }_H\left(r,\sigma ,dr\right)=\left(1-\frac{1}{1+\exp \left(\frac{r-\sigma }{dr\sigma }\right)}\right)$ with the step radius σ, relative exponential width dr.

In the Landau theory of phase transitions, the free energy is expressed in terms of a complex order parameter field $\phi$, where the quantity ${\left|\phi \left(r\right)\right|}^2$ is a measure of the local particle density in analogy to a quantum mechanical wave function.

The generalized Landau ansatz for the partition function is a functional integral [1]

$Z={\displaystyle \int D\phi dV\exp \left(-H\left(\stackrel{\to }{r},\phi ,\beta \right)\right)}$ (23)

where $H\left(\stackrel{\to }{r},\phi ,\beta \right)$ is the thermodynamic Hamiltonian.,

$H\left(\stackrel{\to }{r},\phi ,\beta \right)=E(\phi )+\beta u\left(\stackrel{\to }{r}\right)$, where $E\left(\phi \right)$ is the φ-induced Landau energy, $\beta u\left(\stackrel{\to }{r}\right)$ is the thermal intermolecular potential.

The functional integration $D\phi$ has here a precise mathematical meaning:

$D\phi ={\displaystyle \underset{i}{\sum }\frac{\partial \phi }{\partial c_i}}d{c}_i$ and is perfectly well-defined.

The function variable φ is a function of location $\stackrel{\to }{r}$ and molecular distribution parameters c_i (like molecular diameter σ, average distance λ, correlation length l_c, local distribution periods l_ak) $\phi =\phi \left(r;c_i\right)$.

The volume-integration runs over the location differential $r^2\sin \theta drd\theta d\varphi$.

The actual function φ₀, which yields the valid partition function of a

thermodynamic system, is found by minimization of the free energy

$F=-\frac{1}{\beta }\log Z$ in the parameters c_i in a value range Ω, which yield optimal

parameters c_i0 and the corresponding function ${\phi }_0=\phi \left(r;c_{i0}\right)$

$F\left(\phi \left(c_{i0}\right)\right)=\min \left(F\left(\phi \left(c_i\right)\right);c_i\in \Omega \right)$.

Landau function and the rdf function: a heuristic derivation

We start with the configurational partition function $Z_N={\displaystyle \int d^3{r}_1\cdots d^3{r}_N\exp \left(-\beta U\left({\stackrel{\to }{r}}_1,\cdots ,{\stackrel{\to }{r}}_N\right)\right)}$,

and reformulate it for one particle with spherical symmetry ${\left(Z_N\right)}^{1/N}=z\left(\beta \right)={\displaystyle \int d^{3}rg\left(r\right)\exp \left(-\beta u\left(r\right)\right)}$

where $z\left(\beta \right)$ is the specific (per particle) partition function, $u\left(r\right)=\epsilon u_0\left(r\right)$ is the relative intermolecular potential with characteristic energy ε, $g\left(r\right)$ is the rdf function.

We reformulate it $z\left(\beta \epsilon \right)={\displaystyle \int dVg\left(r\right)\exp \left(-\beta \epsilon u_0\left(r\right)\right)}$.

Now we introduce the finite-range factor $f_{\lambda }=\exp \left(-r/\lambda \right)$ with the average

distance λ, where the specific volume is $v=\frac{V}{N}={\lambda }^3$. The finite-range factor is

approximately $f_{\lambda }=1$ for small distances and large λ in weakly bound systems like gases. $f_{\lambda }$ represents the exponentially decaying local influence within the range λ of an individual particle.

Furthermore, we must take into account the restricted volume with radius σ around the particle, where there are no particles, so we introduce the step function ${\Theta }_H\left(r,\sigma \right)$ into the integral.

Taking all together, we obtain the general Landau form for one particle

$Z={\displaystyle \int D\phi dV\exp \left(-\beta u\left(\stackrel{\to }{r}\right)\right)}\approx {\displaystyle \int \phi \left(r,\sigma ,\lambda ;c_k\right)dV\exp \left(-\beta u\left(\stackrel{\to }{r}\right)\right)}$ (24a)

where the Landau function has the form

$\phi \left(r,\sigma ,\lambda ;c_k\right)={\Theta }_H\left(r,\sigma \right)\exp \left(-\frac{r}{\lambda }\right)g\left(r;c_k\right)$ with parameters $c_k$ (24b)

We know from measurements that the rdf function can be described as a exponentially damped harmonic function, so we make the general ansatz

$g\left(r;c_k\right)=\left(1+\exp \left(-r/l_c\right)\left({\displaystyle \underset{k}{\sum }{\alpha }_k\cos \left(2\pi \frac{r-\sigma }{l_{a,k}}\right)}\right)\right)$ (24c)

Forms of Landau function

For first-order transitions, we make the generalized Landau ansatz for the partition function as a functional integral,

The Landau function depends on parameters $\left(c_k\right)=\left(l_c,{\alpha }_k,l_{a,k},{\alpha }_{0k}\right)$, the

differential is $D\phi ={\displaystyle \sum \frac{\partial \phi }{\partial c_k}}d{c}_k$.

Also, the Boltzmann-exponential contains only the intermolecular potential u, the partition function is

$Z={\displaystyle \int D\phi dV\exp \left(-\beta u\left(\stackrel{\to }{r}\right)\right)}$

The ansatz for the Landau function for solid-fluid-gas transition is the λ-damped rdf function

$\begin{array}{l}\phi \left(r,\sigma ,\lambda ;c_k\right)=\exp \left(-\frac{r}{\lambda }\right){\phi }_{rdf}\left(r,\sigma ;c_k\right)\\ {\phi }_{rdf}\left(r,\sigma ;c_k\right)={\Theta }_H\left(r,\sigma \right)\left(1+\exp \left(-r/l_c\right)\left({\displaystyle \underset{k}{\sum }{\alpha }_k\cos \left(2\pi \frac{r-\sigma }{l_{a,k}}\right)}\right)\right)\end{array}$ (24d)

with the soft-step-up function

${\Theta }_H\left(r,\sigma ,\Delta r\right)=\left(1-\frac{1}{1+\exp \left(\frac{r-\sigma }{\Delta r\sigma }\right)}\right)$

Calculation with this ansatz yields results, which are successfully verified for Lennard-Jones-substances and dipole-substances in chap.10.

Here, we have the typical behavior of first-order phase transitions: there is latent heat, the first derivative of free energy is discontinuous, different phases (e.g. liquid-gas) coexist at transition point.

For second-order transitions, like in magnetic systems, the Landau function has no parameters $\left(c_k\right)=\left\{\right\}$, so the differential is $D\phi =d\phi$.

Also, the partition function contains a φ-generated kinetic energy in the exponential.

$Z={\displaystyle \int D\phi dV\exp \left(-E\left(\phi \right)\right)}$ (25a)

$E\left(\phi \right)$ is the energy generated by $\phi$, and is a generalization of the kinetic energy $\frac{\mu }{2}{\phi }^2$, in Landau’s original ansatz

$E\left(\phi ,\beta \right)=\frac{\mu \left(\beta \right)}{2}{\phi }^2+\frac{\lambda \left(\beta \right)}{4!}{\phi }^4$ (25b)

For magnetic systems, $\phi =M_s=\frac{M}{N}$ specific magnetization, and the thermal potential is $-\beta B\phi$, with interaction $E\left(\phi ,\beta \right)=\frac{\mu \left(\beta \right)}{2}{\phi }^2+\frac{\lambda \left(\beta \right)}{4!}{\phi }^4-\beta \phi B$

Without interaction, the free energy is

$F\left(T,V,N\right)=-\frac{1}{\beta }\log Z\approx \frac{1}{\beta }E\left(\phi ,\beta \right)=\frac{1}{\beta }\left(\frac{\mu \left(\beta \right)}{2}{\phi }^2+\frac{\lambda \left(\beta \right)}{4!}{\phi }^4\right)$,

In this way, we obtain the original Landau ansatz

$F\left(T,\phi \right)=F_0-a\left(T\right){\phi }^2+\frac{b\left(T\right)}{2}{\phi }^4$, $a\left(T\right)=\frac{1}{\beta }\frac{\mu \left(\beta \right)}{2}$, $b\left(T\right)=-\frac{\lambda \left(\beta \right)}{12\beta }$