Particles confined in arbitrary potentials with a class of finite-range repulsiveinteractions

Avanish Kumar, Manas Kulkarni,∗ and Anupam Kundu†

International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru – 560089, India(Dated: October 23, 2020)

In this paper, we develop a large-N field theory for a system of N classical particles in onedimension at thermal equilibrium. The particles are confined by an arbitrary external potential,Vex(x), and repel each other via a class of pairwise interaction potentials Vint(r) (where r is distancebetween a pair of particles) such that Vint ∼ |r|−k when r → 0. We consider the case where everyparticle is interacting with d (finite range parameter) number of particles to its left and right. Dueto the intricate interplay between external confinement, pairwise repulsion and entropy, the densityexhibits markedly distinct behavior in three regimes k > 0, k → 0 and k < 0. From this fieldtheory, we compute analytically the average density profile for large N in these regimes. We showthat the contribution from interaction dominates the collective behaviour for k > 0 and the entropycontribution dominates for k < 0, and both contributes equivalently in the k → 0 limit (finiterange log-gas). Given the fact that these family of systems are of broad relevance, our analyticalfindings are of paramount importance. These results are in excellent agreement with brute-forceMonte-Carlo simulations.

PACS numbers: 75.50.Lk, 64.60.F-, 02.60.Pn

I. INTRODUCTION

Systems of interacting particles confined in exter-nal potentials is ubiquitous in nature. Particularly,pairwise repulsive interactions with power-law diver-gences have taken a special place in physics and math-ematics. There have been several theoretical investi-gations on such systems [1]. Examples include, onedimensional one-component plasma (1dOCP) [2], onedimensional coloumb chain [3], Riesz gas [4–6], Ran-dom Matrix Theory, nuclear physics, mesoscopic trans-port, quantum chaos, number theory [1, 7–9], Calogero-Moser model [10–15], dipolar gas confined to 1d [16–19],screened Coulomb or Yukawa-gas [20, 21] including fi-nance [22] and big-data science [23]. A common featurethat most of the above studies have is that the inter-action among the particles is long-ranged which meansevery particle is interacting with every other particle inthe system. Such interactions have led to developmentsof field theories which have been successfully used to un-derstand various properties like density profiles, numberfluctuations, level-spacing distributions, large deviationsetc; in equilibrium in the large N limit. In the context ofintegrable models, such field theories have also been usedto understand non-equilirbrium features such as shockwaves and solitons [24–26].

In most physical systems, however, interaction betweena pair of particles gets often screened which essentiallymakes the interaction finite-ranged. This naturally raisesthe following question: What are the effects of finite-ranged interactions on the field theory and the conse-quences stemming from it? In this Letter, we precisely

∗ manas.kulkarni@icts.res.in† anupam.kundu@icts.res.in

address this issue by studying a collection of N classi-cal particles with positions {xi} for i = 1, 2, ..., N in aconfining potential Vex(x) in one dimension such thatVex(x) → ∞ as |x| → ∞. Each particle interacts with dparticles on its right and left (if available) and they doso via a repulsive interaction Vint(r) (where r is the dis-tance between a pair of particles) such that Vint ∼ |r|−kwhen r → 0 for k > −k∗ where k∗ is the largest power inthe Taylor series expansion of Vex(r). For k ≤ −k∗, eventhe ground state (obtained from energy minimisation)of the system is unstable because the particles fly off tox = ±∞. It is important to mention that recent cutting-edge developments in experiments has generated a lotof interest in such finite-ranged systems, for e.g., coldatomic gases and ions [27], dipolar bosons [16–18, 28],Rydberg gases [29].

II. MODEL AND PROPERTIES

The total energy of our system is given by

E({xi}) =1

2

N∑

i=1

Vex(xi) +J sgn(k)

2

∑

|i−j|≤dj 6=i

Vint(|xi−xj |)

(1)where J > 0 and d is an integer. Note that the parame-ter d in Eq. (1) determines the number of particles thateach particle is allowed to interact with. For example,by increasing the value of d from 1 to N − 1, one cango from nearest neighbour interaction to all-to-all inter-action scenario. This model is a generalisation of the socalled Riesz gas [4]. Since we are interested in the equi-librium statistical properties of only the position degreesof freedom, the kinetic energy term in the Hamiltonianis omitted.

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For the energy in Eq. (1), the equilibrium joint prob-ability distribution function (PDF) of the positions ofthe particles at finite temperature T = 1/β is givenby P (x1, · · · , xN ) = 1

ZN (β)e−βE[{xi}], where the parti-

tion function ZN (β) =∫ ∏N

i=1 dxi e−βE[{xi}]. While the

confining potential tries to pull all the particles to it’sminimum, the pairwise repulsion as well as the entropytries to spread them apart. Because of this intricatecompetition, it turns out that the particles settle downover a finite region [−`N , `N ] for k > 0 and over thewhole line for k ≤ 0 with an average macroscopic density

〈ρN (x)〉 = N−1∑Ni=1〈δ(x− xi)〉, where 〈. . .〉 denotes an

average with respect to the Boltzmann weight. An im-portant question to ask is: what is the average densityfor large N and how does it depend on T, k and d?

III. KEY FINDINGS

In this paper, we address the question of average den-sity for d ∼ O(1) and find three distinct fascinating sce-narios. We show that, for k > 0 the average density isobtained from a field theory where the interaction termdominates. On the other hand for k < 0 the entropydominates. Remarkably, for k → 0, both interaction andentropy contributes equivalently at finite temperature.

In particular, for an external potential of the polyno-mial form of nth order, Vex(xi) =

∑np=1 apx

pi , we find that

for k > 0, the average density has the following scalingform 〈ρN (x)〉 = `−1

N fk(x/`N ) in the large N limit where

`N = Nk

k+n and

fk(y) = Ad(k) [2µd(k)− Vex(y)]1/k

,

for, |y| ≤ Σ(µd(k)).(2)

The edge of the density Σ(µd(k)) can be obtained fromthe the real zero, closest to the origin, of the equationVex(y) = 2µd(k) and Ad(k) = [2Jζd(k)(k + 1)]−

1k with

ζd(k) =∑dn=1 n

−k. The function µd(k) can then be de-termined by the normalisation condition,

∫ Σ(µd(k))

−Σ(µd(k))

fk(y)dy = 1 . (3)

In order to make sure that all terms in the polynomialcontribute at an equal footing, the coefficients themselves

need to be scaled as ap ∼ Nk

k+n (n−p). It is important tomention that external potentials in the form of polynomi-als are of relevance both experimentally as well as theo-retically [30, 31] and for such potentials we have k∗ = n.Note that, no such finite bound on k exists for thoseVex(x) which have infinite series representations, for e.g.box-like potentials such as Vex(x) = a cosh(bx) [26, 32–36].

In the k < 0 case, for any arbitrary external potentialVex(x), we find that the entropy term dominates to yield

fk(y) = e−βVex(y)/C, for −∞ ≤ y ≤ ∞, (4)

with `N = 1 (no scaling). The normalisation constant Cis fixed by

∫∞−∞ fk(y)dy = 1.

The k → 0 limit turns out to be very interesting andsubtle. To make sense of this limit, we choose Vint(r) =|r|−k. Replacing sgn(k) in Eq. (1) by ±1 for k → 0±, weuse |r|−k ≈ 1− k log |r| and set J = 1/|k|. This up to anoverall additive constant provides

E({xi}) =1

2

N∑

i=1

Vex(xi)−1

2

∑

|i−j|≤dj 6=i

ln |xi − xj | . (5)

We call this system as the finite-range log-gas [37–39].The k → 0 limit can also be taken for some otherchoices of Vint(r) such as 1/| sin(r)|k, 1/|sinh(r)|k, whichyields generalised versions of the finite-range log-gaswhere the interaction term inside the summation be-comes ln | sin(xi − xj)| and ln |sinh(xi − xj)| respectively[1, 7]. For all these cases, it turns out that, contributionsfrom both interaction and entropy appear at the sameorder of N and one finally gets

fk(y) = e−βVex(y)βd+1 /C0, for −∞ ≤ y ≤ ∞ (6)

with `N = 1 and C0 being the normalisation constant.We also performed Monte-Carlo (MC) simulations for

several values of k and find excellent agreement with ouranalytical predictions (see Fig. 1 and 2). In what followswe discuss the derivation of the large N field theory andthe saddle point calculations that lead to our results.

IV. LARGE-N FIELD THEORY

We are interested to compute 〈ρN (x)〉 for large Nwhich is formally given by the following functional in-tegral

〈ρN (x)〉 =

∫D[ρ(z)] P[ρN (z) = ρ(z)] ρ(x), (7)

∀x, where P represents the joint probability density func-tional (JPDF) that ρN (z) = ρ(z), ∀ z ∈ [−∞,∞]. TheJPDF, for large N , can be written as

P[ρN (z) = ρ(z)] =

∫dx1...dxNδ[ρN (z)− ρ(z)]e−βE({xi})

∫dx1...dxNe−βE({xi})

=JN [ρ(z)]e−βEN [ρ(z)]δ

(∫ρ(z)dz − 1

)∫D[h(z)]JN [h(z)]e−βEN [h(z)]δ

(∫h(z)dz − 1

)

where we have assumed that for large N , the energyin Eq. (1) can be expressed as a functional of the

macroscopic density ρN (z) = N−1∑Ni=1 δ(z − xi) i.e.

E({xi}) ≈ EN [ρN (z)]. In fact this is shown explicitlylater [after Eq. (13)]. The combinatorial factor JN [ρ(z)]counts the number of microscopic configurations compat-ible with given macroscopic profile ρ(z). In fact JN [ρ(z)]

3

-2 -1 0 1 20

0.2

0.4

0.6 d=1d=10

-3 -2 -1 0 1 2 30

0.2

0.4

0.6d=1d=10

-3 -2 -1 0 1 2 30

0.2

0.4

0.6 d=5

-2 -1 0 1 20

0.2

0.4

0.6 d=1d=50

-2 -1 0 1 20

0.2

0.4

0.6 d=1d=10

-3 -2 -1 0 1 2 30

0.2

0.4

0.6 d=5

f k(y)

y

k=0.5 k → 0

k=1k=1.5k=-1.5

k=-0.5

(a) (b) (c)

(d) (e) (f)

FIG. 1. Comparison of the densities with Monte-Carlo simu-lation for different values of k and d. The external potential

for all the plots, is Vex(x) = 12

(x4 −N

2kk+2 x2

). The inter-

action potential used in plots (a, c, d, e, f) is Vint(r) = |r|−k

whereas in plot (b) it is Vint(r) = − ln |r|. The solid linesin each plot are from theory and symbols are from numeri-cal simulation. For plots with k > 0 (a, d, e), the theoreti-cal densities are given in Eq. (2). For the Log-gas case (b),k → 0, we compare simulation data with analytical expres-sion in Eq. (6). The plots (c) and (f) on the right columncorresponds to k < 0 where we find Boltzmann-distributiongiven in Eq. (4). Excellent agreement is seen in all cases withno fitting parameters.

is actually the exponential of the entropy associated tomacroscopic density profile ρ(z) [40, 41]

JN [ρ(z)] = e−N∫dz ρ(z) ln ρ(z). (8)

The delta function δ(∫ρ(z)dz − 1

)ensures the normali-

sation of the density functions. Replacing this normalisa-tion constraint by its integral representation

∫dµ2π e−µw =

δ(w) (where the integral is along the imaginary µ axis)we get

P[ρ(z)] =

∫dµ e−SN,µ[ρ(z)]

∫dµ∫D[h(z)]e−SN,µ[h(z)]

, with, (9)

SN,µ[ρ(z)] = βEN [ρ(z)]+N

∫dzρ(z) ln ρ(z)

+ µ

(∫ρ(z)dz − 1

).

(10)

We find (shown later) that the functional SN,µ[ρ(z)], forlarge N grows as Nγk with γk > 1. Hence the parti-tion function in the denominator of the Eq. (9) can beperformed using saddle point method to give

P[ρ(z)] '∫dµ e−(SN,µ[ρ(z)]−SN,µ∗ [ρ∗N (z)]) (11)

where ρ∗N (z) and µ∗ are obtained by minimising the ac-tion in Eq. (10) with respect to ρ(z) and µ i.e solving the

-1 0 10

0.2

0.4

0.6 d=1d=25

-4 -2 0 2 40

0.2

0.4

d=1d=10

-3 -2 -1 0 1 2 30

0.2

0.4

0.6d=1

-10 0 100

0.02

0.04 d=1d=20

-3 -2 -1 0 1 2 30

0.2

0.4

0.6d=1d=10

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8d=1

k=2 k → 0

k → 0k=2

k=-0.5

k=-0.5

y

f k(y)

(a) (b) (c)

(d) (e) (f)

FIG. 2. Demonstration of the validity of our theoretical re-sults in more general cases of interactions as well as externalpotentials. For the plots in the top row, the external po-tential is Vex(x) = x4/2. For the plots in the bottom rowwe have Vex(x) = 1

2Cosh

(x2

), which naturally sets a N in-

dependent length scale i.e. `N ∼ O(1) because it is not inthe form a finite-degree polynomial (diverges exponentiallyat |x| → ∞). The interaction potential used for the plotsin the first column, (a, d) is Vint(r) = 1

|Sinh(r)|k whereas in

the second column, (b, e) it is Vint(r) = ln |Sinh(r)| and

in the third column, (c, f) it is Vint(r) =√r + r5/2

12with

r = |xi − xj |. The solid lines in each plot correspond to ourtheoretical results and the symbols are from numerical sim-ulations. For plot (a), we find that the density is given byEq. (2). In plot (d) we compare our simulation data with the

analytical expression fk(x) = C−1/k(2µ − NCosh(x/2))1/k

with C = 2J(k + 1)ζd(k)Nk+1, where µ is fixed by normal-isation. The solid lines of the plots in the second and thirdcolumns are given in Eqs. (6) and (4), respectively. Onceagain we observe excellent agreement with no fitting param-eters.

following equations

δSN,µ[ρ(z)]

δρ(z)

∣∣∣ρ=ρ∗N

= 0, with

∫dz ρ∗N (z) = 1. (12)

Using the JPDF P from Eq. (11) in Eq. (7) and againperforming a saddle point integration for large N we findthat the average density profile is same as the most prob-able or the typical density profile i.e.

〈ρN (x)〉 = ρ∗N (x). (13)

Next we compute the functional EN [ρ(z)] for the energyfunction given in Eq. (1). To do so we adapt the mainidea of Ref. 5. We first define a smooth function x(s)such that x(i) = xi. This function x(s) becomes uniquein the thermodynamic limit [25] and for a given densityprofile ρ(x), the position function x(i) is given explicitlyby

i = N

∫ x(i)

−∞dz ρ(z). (14)

4

Taking single derivative with respect to x on both sides,we get di/dx = Nρ(x), using which it is easy to see thatfor any smooth function g(xi) of the coordinate xi

∑

i

g(xi) = N

∫dx g(x)ρ(x). (15)

This can be directly applied to the external potentialterm in Eq. (1) to get Eex

N [ρ(x)] = (N/2)∫dx Vex(x)ρ(x).

Expressing the interaction term in terms of the densityprofile ρ(x) is far from obvious and is discussed below.Using Eq. (14), we write the interaction term in Eq. (1)as Eint =

∑∑Vint(|i− j|x′(i) + ...) where we have used

the Taylor series expansion x(j) = x(i)+(j− i)x′(i)+ ....Assuming, x′(i) is small in the large N limit (see ap-pendices) and using Vint(r)|r→0 ∼ |r|−k we get Eint =J ζd(k)sgn(k)

2

∑i(x′(i))−k where we have neglected the

higher order terms in the Taylor series expansion as theyare sub leading. Now inserting x′(i) = 1/(Nρ(x)) andusing Eq. (15) we have

Eint = J ζd(k)sgn(k)Nk+1

∫dxρk+1(x). (16)

Hence the total energy functional EN = Eex +Eint is givenby

EN [ρ(x)] =N

2

∫dx Vex(x)ρ(x)

+ J ζd(k)sgn(k) Nk+1

∫dxρk+1(x).

(17)

Following the same procedure it is possible to show fromEq. (5) that in the k → 0 one gets

EN [ρ(x)] =N

2

∫dx Vex(x)ρ(x) +Nd

∫dxρ(x) ln ρ(x).

(18)

This result can also be obtained directly from Eq. (17) inthe k → 0 limit after setting J = 1/|k|. We now discussthe three regimes separately.

V. DISCUSSIONS

Regime: k > 0: Inserting the above expression ofthe energy functional in Eq. (10), we observe that inthe leading order one can neglect the entropy contri-bution [42]. For an external potential of nth orderpolynomial form, minimizing this action one finds that

〈ρN (x)〉 = ρ∗N (x) = `−1N fk(x/`N ) with `N = N

kk+n and

fk(y) given in Eq. (2). This result is verified numeri-cally. Using this scaling form of the density in the actionin Eq. (10) back, it is easy to see that SN,µ∗ ∼ Nγk

with γk = k(n+1)+nk+n . In fact for k > 1, the formula

in Eq. (2), holds for any d even when d = N − 1 forwhich ζd(k) becomes the usual Riemaan zeta functionζ(k). This happens because, for k > 1, the contribution

from all-to-all interaction comes only at O(N2) which isstill subdominant [5]. It is to be noted that the aboveanalysis fails for very high temperatures of the order

∼ O(N−2kk+n ) when entropy becomes important. In the

special case of a quadratic potential (i.e., n = 2 with,a1 = 0 and a2 = 1), we get the following result for the

support, µd(k) = 18 [Ad(k)B(1 + 1/k, 1 + 1/k)]−

2kk+2 and

Σ(µd(k)) =√

2µd(k).Regime 2: k < 0: In this regime, interestingly, as pair-

wise interaction is of O(N1−|k|), it becomes irrelevant incomparison to the entropy term which is of O(N). There-fore, minimizing the action which now involves only theexternal potential and entropy gives us the usual Boltz-mann distribution in Eq. (4) with `N = 1 for any exter-nal potential. It is noteworthy, that the density profilebecomes independent of the details of the interaction al-though it plays important role to have a description interms of macroscopic particle densities.Regime 3: k → 0: In the case of finite range log-gas, as

can be seen using Eq. (18) in Eq. (10), there is an intri-cate interplay between pairwise interaction and entropybecause they contribute at the same order. Minimiz-ing this action Eq. (10), we get Eq. (6) for any externalpotential. This result was also recently obtained via amicroscopic method [38]. Interaction energy and entropycontributing equivalently has also been observed in loggas with all-to-all interactions [43, 44].

VI. NUMERICAL METHOD AND DETAILS

Our analytical predictions were tested against brute-force MC simulations for N = 501 and β = 1. In oursimulations we collect data after every 10 MC cycles andaverages were performed over around 107 − 108 samplesto compute the particle densities in different cases dis-cussed above. We compare these results with our theo-retical expression in Fig. 1 and Fig. 2 and observe excel-lent agreement in all cases. To make sure that we collectdata after the system has relaxed to equilibrium state,we checked for the equipartition by computing virial

〈xj ∂E({xi})∂xj

〉. The excellent agreement with equiparti-

tion thereby benchmarking our numerics is given in theappendix.

VII. CONCLUSIONS AND OUTLOOK

In this paper, we derive a large-N field theory for asystem of N particles repulsively interacting over a finite-range and confined in arbitrarily external potentials. Wediscuss a family of interaction potentials Vint(r) such thatthey behave as ∼ 1/|r|k for small r. We identify threedistinct regimes depending on the value of k and for eachregimes we derive the action in the large N limit. Min-imising this action provides us explicit expressions of thedensities in arbitrary confining potentials. Our analyti-

5

cal results of densities are in excellent agreement with ourbrute-force numerical simulations. It is pertinent to men-tion that such densities of finite-ranged systems can beexperimentally observed in a broad range of experimentssuch as ions [45, 46], dusty plasma [47] to name a few.This work is of paramount importance since it is essen-tially a starting point for any analysis on a broad classof interacting classical systems. For e.g., if one wantsto study nonlinear hydrodynamics [48, 49], interactingover-damped Langevin particles [50], single-file motion[51], large-deviations [40, 41, 52–54] then writing a large-N field theory is the very beginning step and a correctform of the energy functional is crucial.

Our work paves the path for several future studies suchas non-trivial extension to higher dimensions, extremevalue statistics, level spacing distributions (i.e. statisticsof gap between successive particles) and large deviationfunctions of these externally confined pairwise interactingparticles. Our work acts as a genesis and provides foun-dation for embarking on these exciting directions. Fur-thermore, connections between these models and randommatrix theories remains an open and interesting ques-tion. Finally, it would also be interesting to understandthe crossover from finite-ranged interaction to all-to-allcoupling [5].

VIII. ACKNOWLEDGMENTS

MK would like to acknowledge support from theproject 6004-1 of the Indo-French Centre for the Pro-motion of Advanced Research (IFCPAR), the Ramanu-jan Fellowship SB/S2/RJN-114/2016, the SERB EarlyCareer Research Award ECR/2018/002085 and MatricsGrant (MTR/2019/001101) from the Science and En-gineering Research Board, Department of Science andTechnology, Government of India. AK would like to ac-knowledge support from the project 5604-2 of the Indo-French Centre for the Promotion of Advanced Research(IFCPAR) and the the SERB Early Career ResearchAward ECR/2017/000634 from from the Science and En-gineering Research Board, Department of Science andTechnology, Government of India. We gratefully ac-knowledge Hemanta Kumar G. and ICTS-TIFR high per-formance computing facility.

Appendix A: Continuum approximation for thefinite-range interaction term

In Eq. (1) of the main text, we defined the energy ofa microscopic configuration {xi} as

E({xi}) =1

2

N∑

i=1

Vex(xi)+J sgn(k)

2

∑

|i−j|≤dj 6=i

Vint(|xi−xj |).

(A1)

0 250 500

0.5

1

1.5

0 250 500

0.5

1

1.5

0 250 500

0.5

1

1.5

0 250 500

0.5

1

1.5

0 250 500

0.5

1

1.5

0 250 500

0.5

1

1.5

d=10 d=10 d=5

d=5d=10d=50

k=0.5 k → 0 k=-0.5

k=-1.5k=1k=1.5

(a) (b) (c)

(d) (e) (f)

<x j∂Ε

/∂x j>

j

FIG. 3. Figures above demonstates the virial plots obtainedfrom Eq. (B1) for the corresponsing plots in Fig. (1) of maintext.

0 250 500

0.5

1

1.5

0 250 500

0.5

1

1.5

0 250 500

0.5

1

1.5

0 250 500

0.5

1

1.5

0 250 500

0.5

1

1.5

0 250 500

0.5

1

1.5

d=25k=2

d=20k=2

d=10k → 0

d=10k → 0

d=1k=-0.5

d=1k=-0.5<

x j∂Ε

/∂x j>

j

(a) (b) (c)

(d) (e) (f)

Vint(r) = ln |sinh(r)|<latexit sha1_base64="gIFb1QxsHRmAxn0yA5cpu/7H0gk=">AAACF3icbZDLSsNAFIYnXmu9RV26GWyFuilJXehGKLjRXQV7gSaEyXTaDp1MwsyJUELfwo2v4saFIm5159s4TbvQ1h8Gfr5zDmfOHyaCa3Ccb2tldW19Y7OwVdze2d3btw8OWzpOFWVNGotYdUKimeCSNYGDYJ1EMRKFgrXD0fW03n5gSvNY3sM4YX5EBpL3OSVgUGBXy60g81SEuYRJRZ1deUJiz0wAzrHmcjjlM1YO7JJTdXLhZePOTQnN1QjsL68X0zRiEqggWnddJwE/Iwo4FWxS9FLNEkJHZMC6xkoSMe1n+V0TfGpID/djZZ4EnNPfExmJtB5HoemMCAz1Ym0K/6t1U+hf+hmXSQpM0tmifiowxHgaEu5xxSiIsTGEKm7+iumQKELBRFk0IbiLJy+bVq3qnldrd7VS/XYeRwEdoxNUQS66QHV0gxqoiSh6RM/oFb1ZT9aL9W59zFpXrPnMEfoj6/MHLmyesw==</latexit>

Vint(r) = |sinh(r)|�k<latexit sha1_base64="UCE+qo8RW4tliLVKgjd8dRbBrqg=">AAACGHicbZC7TgJBFIZn8YZ4Qy1tJoIJFuIuFtqYkNhoh4lcEhbJ7DDAhNnZzcxZE7LhMWx8FRsLjbGl822cXSgU/JNJ/nznnJw5vxcKrsG2v63Myura+kZ2M7e1vbO7l98/aOggUpTVaSAC1fKIZoJLVgcOgrVCxYjvCdb0RjdJvfnElOaBfIBxyDo+GUje55SAQd38ebHRjV3lYy5hUlKn167pBpwizeUwYThlj/HZaFLs5gt22U6Fl40zNwU0V62bn7q9gEY+k0AF0brt2CF0YqKAU8EmOTfSLCR0RAasbawkPtOdOD1sgk8M6eF+oMyTgFP6eyImvtZj3zOdPoGhXqwl8L9aO4L+VSfmMoyASTpb1I8EhgAnKeEeV4yCGBtDqOLmr5gOiSIUTJY5E4KzePKyaVTKzkW5cl8pVO/mcWTRETpGJeSgS1RFt6iG6oiiZ/SK3tGH9WK9WZ/W16w1Y81nDtEfWdMfTn+fVQ==</latexit>

Vint(r) =p

|r| + |r|5/2/12<latexit sha1_base64="EB5YO5sbCFxgeWMdSD79ISTsmg8=">AAACJ3icbVDLSgMxFM34tr6qLt0EW0ER6syI6EYpuNFdBfuAzjhk0rQNZjJjckcow/yNG3/FjaAiuvRPTGsXvg6EHM65h+SeMBFcg22/WxOTU9Mzs3PzhYXFpeWV4upaQ8epoqxOYxGrVkg0E1yyOnAQrJUoRqJQsGZ4fTr0m7dMaR7LSxgkzI9IT/IupwSMFBRPyo0g81SEuYR8W+0ce/pGQeaZDGCFR3eOd/EP4So72HPzPcctB8WSXbFHwH+JMyYlNEYtKD55nZimEZNABdG67dgJ+BlRwKlgecFLNUsIvSY91jZUkohpPxvtmeMto3RwN1bmSMAj9XsiI5HWgyg0kxGBvv7tDcX/vHYK3SM/4zJJgUn69VA3FRhiPCwNd7hiFMTAEEIVN3/FtE8UoWCqLZgSnN8r/yUNt+LsV9wLt1Q9H9cxhzbQJtpGDjpEVXSGaqiOKLpDD+gZvVj31qP1ar19jU5Y48w6+gHr4xPCPaSa</latexit>

Vex(x

)=

cosh

(x/2)

/2<latexit sha1_base64="wvxGEqM6mphESE2o4Ph4ZX9D9kE=">AAACCnicbZBPT8IwGMY7/If4b+rRSxVM4ALbPOjFhMSDHjERMIFl6UqBhnZb2s5AFs5e/CpePGiMVz+BN7+NZeyg4JM0+fV53zft+/gRo1JZ1reRW1ldW9/Ibxa2tnd298z9g5YMY4FJE4csFPc+koTRgDQVVYzcR4Ig7jPS9kdXs3r7gQhJw+BOTSLicjQIaJ9ipLTlmcellpd0BYdkPC2PK5cp41AO9a3mVGpOyTOLVtVKBZfBzqAIMjU886vbC3HMSaAwQ1J2bCtSboKEopiRaaEbSxIhPEID0tEYIE6km6SrTOGpdnqwHwp9AgVT9/dEgriUE+7rTo7UUC7WZuZ/tU6s+hduQoMoViTA84f6MYMqhLNcYI8KghWbaEBYUP1XiIdIIKx0egUdgr248jK0nKp9VnVunWL9OosjD47ACSgDG5yDOrgBDdAEGDyCZ/AK3own48V4Nz7mrTkjmzkEf2R8/gBNl5i6</latexit>

Vex(x

)=

x4/2

<latexit sha1_base64="NOAhc3v7punUK+zpp+RmeSxAnLY=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM9gKdVOTKOhGKLjQZQX7gDaGyXTSDp1MwsxEWkLBX3HjQhG3foc7/8Zpm4W2HrhwOOde7r3HjxmVyrK+jdzS8srqWn69sLG5tb1j7u41ZJQITOo4YpFo+UgSRjmpK6oYacWCoNBnpOkPrid+85EISSN+r0YxcUPU4zSgGCkteeZBqeGlHRFCMhyXhydXw4fzU6fkmUWrYk0BF4mdkSLIUPPMr043wklIuMIMSdm2rVi5KRKKYkbGhU4iSYzwAPVIW1OOQiLddHr+GB5rpQuDSOjiCk7V3xMpCqUchb7uDJHqy3lvIv7ntRMVXLop5XGiCMezRUHCoIrgJAvYpYJgxUaaICyovhXiPhIIK51YQYdgz7+8SBpOxT6rOHdOsXqTxZEHh+AIlIENLkAV3IIaqAMMUvAMXsGb8WS8GO/Gx6w1Z2Qz++APjM8f3gSUIg==</latexit>

FIG. 4. Figures above demonstates the virial plots obtainedfrom Eq. (B1) for the corresponsing plots in Fig. (2) of maintext.

where J > 0 and d is an integer. We want to express theinteraction term

Eint =J sgn(k)

2

∑

|i−j|≤dj 6=i

Vint(|xi − xj |)

as a functional of the macroscopic density ρ(z). As notedin the main text, for large N one can define a smoothfunction x(i) such that

i = N

∫ x(i)

−∞dz ρ(z). (A2)

Using this equation, we can write

Eint =J sgn(k)

2

∑

|i−j|≤dj 6=i

Vint

(∣∣∣∞∑

n=1

(i− j)nn!

x[n](i)∣∣∣),

(A3)

where x[n](i) = dnx(i)din . It easy to see (also justified later)

that |x[n+1](i)||x[n](i)| ∼ O(1/N). Hence keeping only the leading

6

order term in the Taylor series expansion in the argumentof Vint we have

Eint ∼J sgn(k)

2

N∑

i=1

∑

|i−j|≤dj 6=i

Vint

(|i− j|x[1](i)

),

∼ J sgn(k)

2

N∑

i=1

∑

|i−j|≤dj 6=i

Vint

( |i− j|Nρ(x(i))

),

using x[1](i) =1

Nρ(x(i))

∼ J sgn(k)

2

N∑

i=1

∑

|i−j|≤dj 6=i

Nkρ(x(i))k

|i− j|k , (A4)

In the last step we used the fact the x[1](i) is small, whichis because of the following. We expect that ρ(x) shouldhave the following scaling form

ρ(x) =1

`Nf

(x

`N

), with, lim

N→∞`NN→ 0. (A5)

Assuming that limit in Eq. (A5) is true we proceed andcompute ρ(x) performing the action minimisation proce-dure explained in the main text and finally check thatthis assumption is indeed true —- thereby making thewhole argument self consistent.Simplifying Eq. (A4) further we get

Eint ∼J sgn(k)

2

N∑

i=1

Nkρ(x(i))k∑

|i−j|≤dj 6=i

1

|i− j|k ,

∼ J sgn(k)

2

N∑

i=1

Nkρ(x(i))kd∑

n=1

1

nk,

∼ J sgn(k)ζd(k)

N∑

i=1

Nkρ(x(i))k,

∼ J sgn(k) ζd(k) Nk+1

∫dxρ(x)k+1, (A6)

using∑

i

g(xi) = N

∫dx g(x)ρ(x)

The above calculation is true for k > −k∗ (see maintext). However for k → 0 (finite-range log-gas) the aboveexpression gets simplified as follows: Setting J = 1/|k|and writing (Nρ(x))k = ek ln(Nρ(x)) and finally taking thek → 0, we obtain

Eint ∼ d N∫dx ρ(x) ln ρ(x) (A7)

upto an overall additive constant where we have useζd(0) = d. Now adding this functional form of Eint[ρ(z)]to Eex[ρ(z)] we get the total energy functional EN [ρ(z)]for k 6= 0

EN [ρ(x)] =N

2

∫dx Vex(x)ρ(x)

+ J ζd(k)sgn(k) Nk+1

∫dxρk+1(x).

(A8)

Following a similar calculation one can show that theenergy functional for k → 0 becomes

EN [ρ(x)] =N

2

∫dx Vex(x)ρ(x) +Nd

∫dxρ(x) ln ρ(x).

(A9)

Inserting these expressions of the energy functionals, inthe expression of the action SN,µ[ρ(z)] below

SN,µ[ρ(z)] = βEN [ρ(z)] +N

∫dzρ(z) ln ρ(z)

+ µ

(∫ρ(z)dz − 1

). (A10)

and minimizing it we get the following saddle point equa-tions

1

2Vex(x) + J (k + 1)ζd(k)sgn(k) Nkρk(x) + µ = 0,

for k > 0,(A11)

1

2Vex(x) + (βd+ 1)[ln ρ(x)− 1] + µ = 0,

for k = 0,(A12)

1

2Vex(x) + [ln ρ(x)− 1] + µ = 0,

for k < 0,(A13)

in the leading order in N . Solving these equations wefind that the saddle point density is given by Eq. (A5)with

`N =

{N

kk+n , for, k > 0

1, for, k ≤ 0(A14)

and

fk(y) =

Ad(k) [2µd(k)− Vex(y)]1/k

, |y| ≤ Σ(µd(k)),

for k > 0,

e−βVex(y)βd+1 /C0, −∞ ≤ y ≤ ∞, for k = 0,

e−βVex(y)/C, −∞ ≤ y ≤ ∞, , for k < 0

as announced in Eqs. (2), (6) and (4) in the main text.Here C and C0 are normalisation constants. This clearlyjustifies the limit in Eq. (A5). It is easy to see that for

7

box-like this limit is trivially true since there is a lengthscale set by the potential itself.

It is important to note that Eq. (A14) holds only whenthe system is stable and there is a notion of a densityprofile, i.e., Eq. (A5). The most suitable way to visualizethis is as follows. If k < k∗, then even for a finite numberof particles N , there is no finite solutions for the parti-cle positions that minimise the energy in Eq. (A1). Allparticles in such a scenario fly away to ±∞ making thediscussion on density to be void.

Appendix B: Virials (equipartition)

To make sure that we collect data after the system hasrelaxed to equilibrium state, we checked for the equipar-

tition by computing virial 〈xj ∂E({xi})∂xj

〉. Below we show

the virials for all the plots in Figs. (1) and (2) in the maintext. The equipartition was tested by checking,

⟨xj∂E({xi})∂xj

⟩= kBT (B1)

Fig. 3 and Fig. 4 shows remarkable agreement therebyvalidating all our numerical results.

Note that for k > 0 when T ∼ O(1), then the fi-nite temperature results match with the density pro-file obtained by minimizing the total energy Eq. (1) inthe main text. In other words, we have N equations∂E({xi})∂xi

= 0, i = 1, 2, ..., N from which can solve for

the N unknows {xmini ; i = 1, 2, ..., N}. Reconstructing

a density function from this (say, by using inverse of in-terparticle distance) will give a density profile which alsowill agree with the one obtained from minimization of theaction described above. This in turn is in perfect agree-ment with brute force finite temperature Monte-Carlo.Needless to mention, this of-course does not encode anyinformation about fluctuations.

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