Coulomb Systems: Thermodynamics, Fluctuations, Large Deviations and Rigidity Joel L. Lebowitz Rutgers University Paris, November 2015 Dedicated to the memory of Janco. Friend, colleague and teacher 1 / 36
The properties of macroscopic matter are almost entirely determined by the Coulomb interactions between electrons and nuclei, satisfying appropriate quantum statistics. While the real world is 3 dimensional it is useful to consider such systems also in other dimensions and as classical systems. The Coulomb interaction between charges e i , e j at positions r i , r j in R d is, with r = | r i − r j | , − e i e j r d = 1 v d ( r ) = − e i e j log( r ) d = 2 (1) + e i e j r 2 − d d ≥ 3 2 / 36
I shall also consider the Jellium or one-component-plasma (OCP) model (introduced by Wigner) in which particles with a positive charge e move in a uniform background of negative charge with density − ρ e . The background produces an external potential proportional to ρ er 2 i ; r i the distance from the center of rotational symmetry. 3 / 36
My concern here will be primarily with effects due to the long range nature of the Coulomb potential. When necessary, we can think of the charges as being smeared out in little balls or having hard cores to take care of the singular contact interactions in d ≥ 2. 4 / 36
To treat such systems via the Gibbs formalism of statistical mechanics we consider globally neutral systems, � N α, j e α = 0 in a sequence of regular domains V j ⊂ R d , such that V j → R d as j → ∞ while the densities N α, j | V j | → ρ α . N j For the OCP | V j | = ρ , the background density in an interval, disc, ball. 5 / 36
In each such box V j the properties of the system will be determined by the canonical measure (density matrix) exp [ − β H j ] µ j = (2) Z ( β, { N j } , V j ) where H j is the Hamiltonian of the system, including both Coulomb and other short range interactions. 6 / 36
We take the“ j → ∞ limit ” of the sequence, f j ≡ − ( β V j ) − 1 log Z ( β, { N j } , V j ) → f � � β, ρ (3) and identify f ( β, ρ ) with the Helmholtz free energy of the macroscopic system. To make this connection with thermodynamics work we have to show that the limit f j → f ( β, ρ ) exists and has the right (convexity) properties. 7 / 36
We take the“ j → ∞ limit ” of the sequence, f j ≡ − ( β V j ) − 1 log Z ( β, { N j } , V j ) → f � � β, ρ (3) and identify f ( β, ρ ) with the Helmholtz free energy of the macroscopic system. To make this connection with thermodynamics work we have to show that the limit f j → f ( β, ρ ) exists and has the right (convexity) properties. This has been proven for both the classical and quantum multi-component and OCP system in d = 1 , 2 , 3. For the OCP the pressure can be negative but we shall not worry about this here. For comprehensive reviews see Martin (1988) and Brydges and Martin (1999). 7 / 36
When there are excess charges Q j in the V j they will go to the surface of the V j and the limit may or may not exist. When it does exist the bulk properties will be the same, in the thermodynamic limit, j → ∞ , as they are for neutral systems. I shall not consider that case here. 8 / 36
Correlations Taking the thermodynamic limit, j → ∞ , we expect to obtain also infinite volume measures µ , at least along sub-sequences. I shall assume the existence of such measures and that they have a unique decomposition into extremal measures which are either translation invariant or periodic. I shall further assume that the latter have correlation functions with decent (at least integrable) clustering properties. 9 / 36
Correlations Taking the thermodynamic limit, j → ∞ , we expect to obtain also infinite volume measures µ , at least along sub-sequences. I shall assume the existence of such measures and that they have a unique decomposition into extremal measures which are either translation invariant or periodic. I shall further assume that the latter have correlation functions with decent (at least integrable) clustering properties. This can be proven for symmetric charges (Frohlich-Park) and at high temperature or low density. (This will be discussed further later). They can also be explicitly determined in some exactly soluble models. 9 / 36
Exact Solutions In d = 1, both the OCP model and the two component model are exactly solvable (Lenard, Baxter, Kunz, . . . ) and one finds that the OCP (but not the two component) system has a periodic structure i.e. a Wigner crystal with period ρ − 1 . In these systems the correlations decay exponentially. 10 / 36
For the OCP in d = 2, one has an exact expression for the correlation functions at one particular temperature β = 2. These correlations have super good clustering properties (Ginibre,Jancovici) with the truncated pair correlation function ρ 2 ( r ) − ρ 2 = − ρ 2 e − πρ r 2 , r = | r 1 − r 2 | (4) Higher order truncated correlations also decay like ∼ e − γ D 2 , D the distance between groups of particles. 11 / 36
Fluctuations To fluctuate is normal and in most cases fluctuations are themselves normal, by which I mean that in a region Λ with volume | Λ | , they grow like the square root of | Λ | as in a Poisson process (or faster as at critical points). There are however many very interesting cases where the fluctuations are subnormal. This includes local charge fluctuations in globally neutral macroscopic systems, the case I shall now discuss. 12 / 36
To get a feeling for what such fluctuations might look like we note that in many situations, such as those involving fluids at low and moderate temperatures, we usually consider macroscopic systems as made up of neutral atoms or molecules interacting via effective short range Lennard-Jones type potentials. In such cases, the fluctuations in the net charge Q Λ in a region Λ will be due entirely to the surface of Λ cutting these entities in a “random” way. < Q 2 Λ > may then be expected to be proportional to the surface area of Λ. 13 / 36
The question naturally arises as to whether this type of behavior is indeed a consequence, in some or all situations, of the true Coulomb interactions. In particular, is it true for charge fluctuations in plasmas, molten salts, metals, etc., where bare Coulomb interactions are part of the effective Hamiltonian? 14 / 36
The question naturally arises as to whether this type of behavior is indeed a consequence, in some or all situations, of the true Coulomb interactions. In particular, is it true for charge fluctuations in plasmas, molten salts, metals, etc., where bare Coulomb interactions are part of the effective Hamiltonian? To simplify matters I shall consider the classical OCP (with e = 1) whose structure is of interest also in other contexts, such as the distribution of eigenvalues of random matrices. I will indicate the difference with multi-component systems when relevant. 14 / 36
Now, while for systems with short range interactions one can prove (Ginibre) that the variance in particle number N Λ in a region Λ ⊂ R d grows at least as fast as the volume | Λ | V Λ = < ( N Λ − < N Λ > ) 2 > ≥ c | Λ | , c > 0 , (5) this does not hold for Coulomb interactions. Fluctuations in the charge Q Λ , which for the OCP is the same as fluctuations in N Λ with < N Λ > = ρ | Λ | , will, as already noted, only grow as the surface area < Q 2 Λ > ∼ | ∂ Λ | . This is in fact what one can prove, under reasonable assumptions on clustering. 15 / 36
To see how this comes about we note that the variance V Λ is expressible in terms of the pair correlation function of the infinite system. For a translation invariant system we have, � � V Λ = d r 1 d r 2 G ( r 1 − r 2 ) Λ Λ � � = | Λ | R d G ( r ) d r − R d G ( r ) α Λ ( r ) d r , where �� � − ρ 2 , G ( r 1 − r 2 ) = δ ( r 1 − x i ) δ ( r 2 − x j ) i , j = ρδ ( r 1 − r 2 ) + ρ 2 ( r 1 − r 2 ) − ρ 2 , � α Λ ( r ) = χ Λ ( r + r 1 )[1 − χ Λ ( r 1 )] d r 1 � 1 y ∈ Λ χ Λ ( y ) = 0 y / ∈ Λ 16 / 36
This is modified in a simple way for a periodic system. For charge fluctuations in multi-charge systems G ( r ) corresponds to the charge-charge correlations. 17 / 36
When Λ ↑ R d in a self similar way α Λ will grow like the surface area | ∂ Λ | ∼ | Λ | ( d − 1) / d with | ∂ Λ | = 2 for d = 1. Averaging α Λ ( r ) / | ∂ Λ | over rotations we obtain α Λ ( r ) lim | ∂ Λ | = α d | r | , | Λ |→∞ where 1 / 2 d = 1 1 /π d = 2 α d = . . . 18 / 36
In Coulomb systems lim 1 � | Λ |V Λ = R d G ( r ) d r = 0 , (6) due to Debye screening. This is known as the “first sum rule”. Systems satisfying (6) are also known as superhomogeneous. 19 / 36
In Coulomb systems lim 1 � | Λ |V Λ = R d G ( r ) d r = 0 , (6) due to Debye screening. This is known as the “first sum rule”. Systems satisfying (6) are also known as superhomogeneous. We then have, for systems satisfying (6), � ∞ V Λ r d G ( r ) dr , | ∂ Λ | → − α d (7) 0 where we have sphericalized G . Equation (7) is called the Stillinger-Lovett relation. When (6) holds but (7) is infinite the variance will grow faster than the surface area but slower than the volume. 19 / 36
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