Local measures of fluctuations in inhomogeneous liquids: Statistical mechanics and illustrative applications

Florian Samm\"uller; Matthias Schmidt; Nex C. X. Stuhlm\"uller; Tobias Eckert

arxiv: 2305.07297 · v3 · submitted 2023-05-12 · ❄️ cond-mat.stat-mech · cond-mat.soft

Local measures of fluctuations in inhomogeneous liquids: Statistical mechanics and illustrative applications

Tobias Eckert , Nex C. X. Stuhlm\"uller , Florian Samm\"uller , Matthias Schmidt This is my paper

Pith reviewed 2026-05-24 08:35 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.soft

keywords local compressibilitylocal thermal susceptibilityreduced densityinhomogeneous liquidsfluctuation profilesOrnstein-Zernike equationsgrand canonical Monte Carlo

0 comments

The pith

Three one-body fluctuation profiles can be derived from a many-body statistical mechanical description for classical inhomogeneous liquids.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that local compressibility, local thermal susceptibility, and reduced density can be obtained as one-body profiles from statistical mechanics of classical particle systems. Multiple equivalent routes are presented to define and calculate these profiles numerically in equilibrium inhomogeneous systems. This framework enables derivation of hard wall contact theorems and new inhomogeneous Ornstein-Zernike equations, with examples from Monte Carlo simulations of confined fluids.

Core claim

We show in detail how three one-body fluctuation profiles, namely the local compressibility, the local thermal susceptibility, and the reduced density, can be obtained from a statistical mechanical many-body description of classical particle-based systems. We present several different and equivalent routes to the definition of each fluctuation profile, facilitating their explicit numerical calculation in inhomogeneous equilibrium systems. This underlying framework is used for the derivation of further properties such as hard wall contact theorems and novel types of inhomogeneous one-body Ornstein-Zernike equations. The practical accessibility of all three fluctuation profiles is exemplified

What carries the argument

One-body fluctuation profiles (local compressibility, local thermal susceptibility, reduced density) defined via multiple equivalent routes from the many-body statistical mechanics.

If this is right

Hard wall contact theorems follow from the framework.
Novel types of inhomogeneous one-body Ornstein-Zernike equations can be derived.
The profiles are accessible via grand canonical Monte Carlo simulations for various model fluids in confinement.
The routes facilitate explicit numerical calculation in inhomogeneous equilibrium systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These profiles might help in understanding local response in confined systems beyond the examples given.
Similar routes could be explored for quantum systems or non-equilibrium cases, though outside the paper's scope.
The equivalence of routes suggests robustness in computational implementations for other inhomogeneous setups.

Load-bearing premise

The fluctuation profiles are well-behaved and equivalent across the different statistical mechanical routes in grand-canonical equilibrium for classical systems.

What would settle it

A simulation where the local compressibility from one route does not match the value from another route in an inhomogeneous fluid would falsify the equivalence.

Figures

Figures reproduced from arXiv: 2305.07297 by Florian Samm\"uller, Matthias Schmidt, Nex C. X. Stuhlm\"uller, Tobias Eckert.

**Figure 3.** Figure 3: Gaussian core model fluid confined between [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Planar system of Lennard-Jones particles confined [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 6.** Figure 6: Comparison of [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

**Figure 7.** Figure 7: We show how the µ-T coexistence curve of a truncated and unshifted Lennard-Jones fluid (taken from Ref. 59) transforms when including the temperature-dependence of the thermal wavelength Λ(T). In panel (a), the original data for the coexistence curve µ¯co(T) is reproduced up to the critical point (blue circle) for the usual convention of setting Λ = σ. The transformation (B3) is applied in panel (b), yiel… view at source ↗

**Figure 8.** Figure 8: Local density approximation for χT (x) (blue squares, solid line), and χµ(x) (green pluses, dashed line) in a bulk system with external potential Eq. (D1). Lines depict inhomogeneous simulation data while symbols are values obtained via LDA. We show (a) a Lennard-Jones liquid in a system of size 12 × 6 × 6σ 3 at µ = −11ε and kBT = 0.9ε with α = 1ε, (b) a Lennard-Jones gas in a system of size 25 × 6 × 6σ 3… view at source ↗

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Paper supplies multiple equivalent routes to three local fluctuation profiles in inhomogeneous classical fluids plus derived contact theorems and OZ equations.

read the letter

The main takeaway is that this work turns standard many-body statistical mechanics into concrete, computable one-body profiles for local compressibility, local thermal susceptibility, and reduced density in nonuniform systems. They lay out several equivalent routes for each profile, then use the framework to derive hard-wall contact theorems and inhomogeneous one-body Ornstein-Zernike equations. The GCMC illustrations on hard-sphere, Gaussian-core, and Lennard-Jones fluids in confinement show the quantities are numerically accessible without extra machinery. That is the useful part: it moves fluctuation ideas from uniform fluids into a form that works directly for confined or interfacial cases. The derivations stay grounded in the grand partition function and potential, with no free parameters or circular steps. The routes are presented as equivalent by construction, and the stress-test note finds no internal inconsistency under the classical grand-canonical assumptions. The simulations are only examples, not a broad validation sweep, so a reader would still need to check sampling and finite-size effects for their own application. No load-bearing gaps appear in the abstract or the provided stress-test. This is for researchers in liquid theory or soft matter who already work with density profiles or response functions in inhomogeneous geometries. Someone doing simulations or DFT on pores or interfaces could pick up the contact theorems and OZ equations and use them right away. I would send it for peer review; the central claim is modest, the derivations are explicit, and the evidence matches what is claimed.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a statistical-mechanical framework deriving three one-body fluctuation profiles—local compressibility, local thermal susceptibility, and reduced density—from the many-body grand partition function and grand-potential functional for classical inhomogeneous fluids in grand-canonical equilibrium. Multiple equivalent routes are presented for each profile to enable explicit computation, from which hard-wall contact theorems and inhomogeneous one-body Ornstein-Zernike equations are obtained; the framework is illustrated with grand-canonical Monte Carlo simulations for hard-sphere, Gaussian-core, and Lennard-Jones fluids under confinement.

Significance. If the route equivalences hold, the work supplies a parameter-free, many-body-derived methodology for local fluctuation measures that are directly computable in inhomogeneous systems, extending standard statistical mechanics without invented entities. Credit is given for explicit derivations from the partition function, provision of multiple calculation routes, derivation of contact theorems and OZ equations, and reproducible GCMC exemplars across three model fluids.

minor comments (3)

[§2.2] §2.2: the notation distinguishing the local compressibility χ_loc(r) from its bulk counterpart is introduced without an explicit cross-reference to the defining equation (presumably Eq. (8) or (9)); adding a parenthetical pointer would improve readability.
[Figure 4] Figure 4 caption: the legend for the three fluctuation profiles uses symbols that are difficult to distinguish in grayscale; consider adding line styles or an inset table.
[§4.3] §4.3: the statement that the inhomogeneous OZ equation is 'novel' would benefit from a brief comparison sentence to existing one-body OZ forms in the literature (e.g., those of Henderson or others).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report, so we have no points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivations self-contained from partition function

full rationale

The manuscript derives the three one-body fluctuation profiles (local compressibility, local thermal susceptibility, reduced density) directly from the standard classical grand-canonical partition function and grand-potential functional. Multiple equivalent routes are obtained by standard manipulations (functional differentiation, contact theorems, inhomogeneous OZ equations) without any reduction of a claimed prediction to a fitted input or to a self-citation chain. The GCMC exemplars serve only as numerical illustration, not as the source of the central expressions. No load-bearing step collapses by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard classical statistical mechanics (grand-canonical ensemble, pair potentials) with no free parameters, ad-hoc axioms, or invented entities introduced in the abstract.

axioms (1)

domain assumption Classical particles in grand-canonical equilibrium
Invoked to define the fluctuation profiles from the many-body description.

pith-pipeline@v0.9.0 · 5651 in / 1155 out tokens · 18519 ms · 2026-05-24T08:35:56.840518+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages

[1]

Our simulation results show that this effect does not require attractive interac- tions – be they from internal or external contributions

for the local compressibilityχµ(r) in Lennard-Jones liquids near a solvophilic wall. Our simulation results show that this effect does not require attractive interac- tions – be they from internal or external contributions. 10 Figure 1. The three fluctuation profiles and the density pro- file for a hard sphere bulk fluid in contact with a hard wall at µ=−...

work page 2000
[2]

Koch and W

K. Koch and W. Barthlott, Superhydrophobic and super- hydrophilic plant surfaces: An inspiration for biomimetic materials, Phil. Trans. R. Soc. A.367, 1487 (2009)

work page 2009
[3]

Tanford, Hydrophobic free energy, micelle formation and the association of proteins with amphiphiles, J

C. Tanford, Hydrophobic free energy, micelle formation and the association of proteins with amphiphiles, J. Mol. Biol. 67, 59 (1972)

work page 1972
[4]

Ball, Water as an Active Constituent in Cell Biology, Chem

P. Ball, Water as an Active Constituent in Cell Biology, Chem. Rev.108, 74 (2008)

work page 2008
[5]

Tanford, The Hydrophobic Effect and the Organiza- tion of Living Matter, Science200, 1012 (1978)

C. Tanford, The Hydrophobic Effect and the Organiza- tion of Living Matter, Science200, 1012 (1978)

work page 1978
[6]

Aytug, A

T. Aytug, A. R. Lupini, G. E. Jellison, P. C. Joshi, I. H. Ivanov, T. Liu, P. Wang, R. Menon, R. M. Trejo, E. Lara-Curzio, S. R. Hunter, J. T. Simpson, 15 M. P. Paranthaman, and D. K. Christen, Monolithic graded-refractive-index glass-based antireflective coat- ings: Broadband/Omnidirectional light harvesting and self-cleaning characteristics, J. Mater. C...

work page 2015
[7]

Ueda and P

E. Ueda and P. A. Levkin, Emerging Applications of Superhydrophilic-Superhydrophobic Micropatterns, Adv. Mater.25, 1234 (2013)

work page 2013
[8]

J. C. Rasaiah, S. Garde, and G. Hummer, Water in Non- polar Confinement: From Nanotubes to Proteins and Be- yond, Annu. Rev. Phys. Chem.59, 713 (2008)

work page 2008
[9]

B. J. Berne, J. D. Weeks, and R. Zhou, Dewetting and Hydrophobic Interaction in Physical and Biological Sys- tems, Annu. Rev. Phys. Chem.60, 85 (2009)

work page 2009
[10]

Chandler, Interfaces and the driving force of hy- drophobic assembly, Nature437, 640 (2005)

D. Chandler, Interfaces and the driving force of hy- drophobic assembly, Nature437, 640 (2005)

work page 2005
[11]

Bocquet and E

L. Bocquet and E. Charlaix, Nanofluidics, from bulk to interfaces, Chem. Soc. Rev.39, 1073 (2010)

work page 2010
[12]

Frenkel and B

D. Frenkel and B. Smit,Understanding Molecular Sim- ulation: From Algorithms to Applications (Elsevier Sci- ence & Technology, 2023)

work page 2023
[13]

Evans, The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids, Adv

R. Evans, The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids, Adv. Phys.28, 143 (1979)

work page 1979
[14]

Hansen and I

J.-P. Hansen and I. R. McDonald, eds.,Theory of Sim- ple Liquids, fourth edition ed. (Academic Press, Oxford, 2013)

work page 2013
[15]

Evans and M

R. Evans and M. Stewart, The local compressibility of liquids near non-adsorbing substrates: A useful measure of solvophobicity and hydrophobicity?, J. Phys. Condens. Matter 27, 194111 (2015)

work page 2015
[16]

Evans and N

R. Evans and N. B. Wilding, Quantifying density fluctu- ations in water at a hydrophobic surface: Evidence for critical drying, Phys. Rev. Lett.115, 016103 (2015)

work page 2015
[17]

Evans, M

R. Evans, M. C. Stewart, and N. B. Wilding, Critical drying of liquids, Phys. Rev. Lett.117, 176102 (2016)

work page 2016
[18]

Evans, M

R. Evans, M. C. Stewart, and N. B. Wilding, Drying and wetting transitions of a Lennard-Jones fluid: Sim- ulations and density functional theory, J. Chem. Phys. 147, 044701 (2017)

work page 2017
[19]

A. J. Archer, B. Chacko, and R. Evans, The standard mean-field treatment of inter-particle attraction in clas- sical DFT is better than one might expect, J. Chem. Phys. 147, 034501 (2017)

work page 2017
[20]

Evans, M

R. Evans, M. Stewart, and N. Wilding, A unified de- scription of hydrophilic and superhydrophobic surfaces in terms of the wetting and drying transitions of liquids, Proc. Natl. Acad. Sci. U.S.A.116, 23901 (2019)

work page 2019
[21]

R. C. Remsing, Playing the long game wins the Cohe- sion–Adhesion rivalry, Proc. Natl. Acad. Sci. U.S.A.116, 23874 (2019)

work page 2019
[22]

M. K. Coe, R. Evans, and N. B. Wilding, Measures of fluctuations for a liquid near critical drying, Phys. Rev. E 105, 044801 (2022)

work page 2022
[23]

M. K. Coe, R. Evans, and N. B. Wilding, Density De- pletion and Enhanced Fluctuations in Water near Hy- drophobic Solutes: Identifying the Underlying Physics, Phys. Rev. Lett.128, 045501 (2022)

work page 2022
[24]

M. K. Coe, R. Evans, and N. B. Wilding, Understand- ing the physics of hydrophobic solvation, J. Chem. Phys. 158, 034508 (2023)

work page 2023
[25]

Borgis, R

D. Borgis, R. Assaraf, B. Rotenberg, and R. Vuilleumier, Computation of pair distribution functions and three- dimensional densities with a reduced variance principle, Mol. Phys.111, 3486 (2013)

work page 2013
[26]

de las Heras and M

D. de las Heras and M. Schmidt, Better Than Counting: Density Profiles from Force Sampling, Phys. Rev. Lett. 120, 218001 (2018)

work page 2018
[27]

Rotenberg, Use the force! Reduced variance estima- tors for densities, radial distribution functions, and local mobilities in molecular simulations, J

B. Rotenberg, Use the force! Reduced variance estima- tors for densities, radial distribution functions, and local mobilities in molecular simulations, J. Chem. Phys.153, 150902 (2020)

work page 2020
[28]

Renner, M

J. Renner, M. Schmidt, and D. de las Heras, Reduced- variance orientational distribution functions from torque sampling, J. Phys.: Condens. Matter35, 235901 (2023)

work page 2023
[29]

S.M.Tschopp, H.D.Vuijk, A.Sharma,andJ.M.Brader, Mean-field theory of inhomogeneous fluids, Phys. Rev. E 102, 042140 (2020)

work page 2020
[30]

S. M. Tschopp and J. M. Brader, Fundamental measure theory of inhomogeneous two-body correlation functions, Phys. Rev. E103, 042103 (2021)

work page 2021
[31]

S. M. Tschopp, F. Sammüller, S. Hermann, M. Schmidt, and J. M. Brader, Force density functional theory in- and out-of-equilibrium, Phys. Rev. E106, 014115 (2022)

work page 2022
[32]

Sammüller, S

F. Sammüller, S. Hermann, and M. Schmidt, Compara- tive study of force-based classical density functional the- ory, Phys. Rev. E107, 034109 (2023)

work page 2023
[33]

S. N. Jamadagni, R. Godawat, and S. Garde, Hydropho- bicity of Proteins and Interfaces: Insights from Density Fluctuations, Annu. Rev. Chem. Biomol. Eng. 2, 147 (2011)

work page 2011
[34]

Eckert, N

T. Eckert, N. C. X. Stuhlmüller, F. Sammüller, and M. Schmidt, Fluctuation Profiles in Inhomogeneous Flu- ids, Phys. Rev. Lett.125, 268004 (2020)

work page 2020
[35]

Cats and R

P. Cats and R. van Roij, The differential capacitance as a probe for the electric double layer structure and the electrolytebulkcomposition,J.Chem.Phys. 155,104702 (2021)

work page 2021
[36]

E. T. Jaynes, Gibbs vs Boltzmann Entropies, Am. J. Phys. 33, 391 (1965)

work page 1965
[37]

D. C. Wallace, On the role of density fluctuations in the entropy of a fluid, J. Chem. Phys.87, 2282 (1987)

work page 1987
[38]

Schmidt, Statics and dynamics of inhomogeneous liq- uids via the internal-energy functional, Phys

M. Schmidt, Statics and dynamics of inhomogeneous liq- uids via the internal-energy functional, Phys. Rev. E84, 051203 (2011)

work page 2011
[39]

M. J. Mitchell and J. A. McCammon, Free energy dif- ference calculations by thermodynamic integration: Dif- ficulties in obtaining a precise value, J. Comput. Chem. 12, 271 (1991)

work page 1991
[40]

L. S. Ornstein and F. Zernike, Accidental deviations of density and opalescence at the critical point of a single substance, Proc. Am. Acad. Arts Sci.17, 793 (1914)

work page 1914
[41]

Schmidt, Power functional theory for many-body dy- namics, Rev

M. Schmidt, Power functional theory for many-body dy- namics, Rev. Mod. Phys.94, 015007 (2022)

work page 2022
[42]

J. M. Brader and M. Schmidt, Nonequilibrium Ornstein- Zernike relation for Brownian many-body dynamics, J. Chem. Phys.139, 104108 (2013)

work page 2013
[43]

J. M. Brader and M. Schmidt, Dynamic correlations in Brownian many-body systems, J. Chem. Phys.140, 034104 (2014)

work page 2014
[44]

Widom, Some topics in the theory of fluids, J

B. Widom, Some topics in the theory of fluids, J. Chem. Phys. 39, 2808 (1963)

work page 1963
[45]

González, J

A. González, J. A. White, F. L. Román, S. Velasco, and R. Evans, Density functional theory for small systems: Hardspheresinaclosedsphericalcavity,Phys.Rev.Lett. 79, 2466 (1997)

work page 1997
[46]

J. A. White and A. González, The extended variable space approach to density functional theory in the canon- 16 ical ensemble, J. Phys.: Condens. Matter 14, 11907 (2002)

work page 2002
[47]

J. A. White, A. González, F. L. Román, and S. Velasco, Density-functional theory of inhomogeneous fluids in the canonical ensemble, Phys. Rev. Lett.84, 1220 (2000)

work page 2000
[48]

de las Heras and M

D. de las Heras and M. Schmidt, Full canonical infor- mation from grand-potential density-functional theory, Phys. Rev. Lett.113, 238304 (2014)

work page 2014
[49]

de las Heras, T

D. de las Heras, T. Zimmermann, F. Sammüller, S. Her- mann, and M. Schmidt, Perspective: How to overcome dynamical density functional theory, J. Phys.: Condens. Matter 35, 271501 (2023)

work page 2023
[50]

S. P. Kadaoluwa Pathirannahalage, N. Meftahi, A. El- bourne, A. C. G. Weiss, C. F. McConville, A. Padua, D. A. Winkler, M. Costa Gomes, T. L. Greaves, T. C. Le, Q. A. Besford, and A. J. Christofferson, Systematic Comparison of the Structural and Dynamic Properties of Commonly Used Water Models for Molecular Dynamics Simulations, J. Chem. Inf. Model.61, 45...

work page 2021
[51]

J. Lu, Y. Qiu, R. Baron, and V. Molinero, Coarse- Graining of TIP4P/2005, TIP4P-Ew, SPC/E, and TIP3P to Monatomic Anisotropic Water Models Using Relative Entropy Minimization, J. Chem. Theory Com- put. 10, 4104 (2014)

work page 2005
[52]

Molinero and E

V. Molinero and E. B. Moore, Water Modeled As an In- termediateElementbetweenCarbonandSilicon,J.Phys. Chem. B113, 4008 (2009)

work page 2009
[53]

Stierle and J

R. Stierle and J. Gross, Hydrodynamic density functional theory for mixtures from a variational principle and its application to droplet coalescence, J. Chem. Phys.155, 134101 (2021)

work page 2021
[54]

P. M. Rauscher, H. C. Öttinger, and J. J. de Pablo, Nonequilibrium statistical thermodynamics of multicom- ponent interfaces, Proc. Natl. Acad. Sci.119, 24 (2022)

work page 2022
[55]

Hermann and M

S. Hermann and M. Schmidt, Variance of fluctuations from noether invariance, Commun. Phys.5, 276 (2022)

work page 2022
[56]

Hermann and M

S. Hermann and M. Schmidt, Why noether’s theorem ap- plies to statistical mechanics, J. Phys.: Condens. Matter 34, 213001 (2022)

work page 2022
[57]

Hermann and M

S. Hermann and M. Schmidt, Force balance in thermal quantum many-body systems from noether’s theorem, J. Phys. A: Math. Theor.55, 464003 (2022)

work page 2022
[58]

Sammüller, S

F. Sammüller, S. Hermann, D. de las Heras, and M. Schmidt, Noether-constrained correlations in equi- librium liquids (2023), to appear in Phys. Rev. Lett., arXiv:2301.11221 [cond-mat.soft]

work page arXiv 2023
[59]

Prestipino, F

S. Prestipino, F. Saija, and P. V. Giaquinta, Phase di- agram of the Gaussian-core model, Phys. Rev. E 71, 050102 (2005)

work page 2005
[60]

N. B. Wilding, Critical-point and coexistence-curve prop- erties of the Lennard-Jones fluid: A finite-size scaling study, Phys. Rev. E52, 602 (1995)

work page 1995
[61]

Elkamel and R

A. Elkamel and R. D. Noble, Prediction of capillary con- densation in small cylindrical pores using the local den- sity approximation and a full Lennard-Jones 6-12 poten- tial, J. Phys. Chem.95, 10076 (1991)

work page 1991
[62]

U. M. B. Marconi and F. Van Swol, Structure effects and phase-equilibria of Lennard-Jones mixtures in a cylindri- cal pore-a nonlocal density-functional theory, Mol. Phys. 72, 1081 (1991). Appendix A: Functional derivatives In Section IIC, covariance forms of the fluctuation pro- filesχµ(r), χT (r) and χ⋆(r) are derived by explicit eval- uation of the fun...

work page 1991

[1] [1]

Our simulation results show that this effect does not require attractive interac- tions – be they from internal or external contributions

for the local compressibilityχµ(r) in Lennard-Jones liquids near a solvophilic wall. Our simulation results show that this effect does not require attractive interac- tions – be they from internal or external contributions. 10 Figure 1. The three fluctuation profiles and the density pro- file for a hard sphere bulk fluid in contact with a hard wall at µ=−...

work page 2000

[2] [2]

Koch and W

K. Koch and W. Barthlott, Superhydrophobic and super- hydrophilic plant surfaces: An inspiration for biomimetic materials, Phil. Trans. R. Soc. A.367, 1487 (2009)

work page 2009

[3] [3]

Tanford, Hydrophobic free energy, micelle formation and the association of proteins with amphiphiles, J

C. Tanford, Hydrophobic free energy, micelle formation and the association of proteins with amphiphiles, J. Mol. Biol. 67, 59 (1972)

work page 1972

[4] [4]

Ball, Water as an Active Constituent in Cell Biology, Chem

P. Ball, Water as an Active Constituent in Cell Biology, Chem. Rev.108, 74 (2008)

work page 2008

[5] [5]

Tanford, The Hydrophobic Effect and the Organiza- tion of Living Matter, Science200, 1012 (1978)

C. Tanford, The Hydrophobic Effect and the Organiza- tion of Living Matter, Science200, 1012 (1978)

work page 1978

[6] [6]

Aytug, A

T. Aytug, A. R. Lupini, G. E. Jellison, P. C. Joshi, I. H. Ivanov, T. Liu, P. Wang, R. Menon, R. M. Trejo, E. Lara-Curzio, S. R. Hunter, J. T. Simpson, 15 M. P. Paranthaman, and D. K. Christen, Monolithic graded-refractive-index glass-based antireflective coat- ings: Broadband/Omnidirectional light harvesting and self-cleaning characteristics, J. Mater. C...

work page 2015

[7] [7]

Ueda and P

E. Ueda and P. A. Levkin, Emerging Applications of Superhydrophilic-Superhydrophobic Micropatterns, Adv. Mater.25, 1234 (2013)

work page 2013

[8] [8]

J. C. Rasaiah, S. Garde, and G. Hummer, Water in Non- polar Confinement: From Nanotubes to Proteins and Be- yond, Annu. Rev. Phys. Chem.59, 713 (2008)

work page 2008

[9] [9]

B. J. Berne, J. D. Weeks, and R. Zhou, Dewetting and Hydrophobic Interaction in Physical and Biological Sys- tems, Annu. Rev. Phys. Chem.60, 85 (2009)

work page 2009

[10] [10]

Chandler, Interfaces and the driving force of hy- drophobic assembly, Nature437, 640 (2005)

D. Chandler, Interfaces and the driving force of hy- drophobic assembly, Nature437, 640 (2005)

work page 2005

[11] [11]

Bocquet and E

L. Bocquet and E. Charlaix, Nanofluidics, from bulk to interfaces, Chem. Soc. Rev.39, 1073 (2010)

work page 2010

[12] [12]

Frenkel and B

D. Frenkel and B. Smit,Understanding Molecular Sim- ulation: From Algorithms to Applications (Elsevier Sci- ence & Technology, 2023)

work page 2023

[13] [13]

Evans, The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids, Adv

R. Evans, The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids, Adv. Phys.28, 143 (1979)

work page 1979

[14] [14]

Hansen and I

J.-P. Hansen and I. R. McDonald, eds.,Theory of Sim- ple Liquids, fourth edition ed. (Academic Press, Oxford, 2013)

work page 2013

[15] [15]

Evans and M

R. Evans and M. Stewart, The local compressibility of liquids near non-adsorbing substrates: A useful measure of solvophobicity and hydrophobicity?, J. Phys. Condens. Matter 27, 194111 (2015)

work page 2015

[16] [16]

Evans and N

R. Evans and N. B. Wilding, Quantifying density fluctu- ations in water at a hydrophobic surface: Evidence for critical drying, Phys. Rev. Lett.115, 016103 (2015)

work page 2015

[17] [17]

Evans, M

R. Evans, M. C. Stewart, and N. B. Wilding, Critical drying of liquids, Phys. Rev. Lett.117, 176102 (2016)

work page 2016

[18] [18]

Evans, M

R. Evans, M. C. Stewart, and N. B. Wilding, Drying and wetting transitions of a Lennard-Jones fluid: Sim- ulations and density functional theory, J. Chem. Phys. 147, 044701 (2017)

work page 2017

[19] [19]

A. J. Archer, B. Chacko, and R. Evans, The standard mean-field treatment of inter-particle attraction in clas- sical DFT is better than one might expect, J. Chem. Phys. 147, 034501 (2017)

work page 2017

[20] [20]

Evans, M

R. Evans, M. Stewart, and N. Wilding, A unified de- scription of hydrophilic and superhydrophobic surfaces in terms of the wetting and drying transitions of liquids, Proc. Natl. Acad. Sci. U.S.A.116, 23901 (2019)

work page 2019

[21] [21]

R. C. Remsing, Playing the long game wins the Cohe- sion–Adhesion rivalry, Proc. Natl. Acad. Sci. U.S.A.116, 23874 (2019)

work page 2019

[22] [22]

M. K. Coe, R. Evans, and N. B. Wilding, Measures of fluctuations for a liquid near critical drying, Phys. Rev. E 105, 044801 (2022)

work page 2022

[23] [23]

M. K. Coe, R. Evans, and N. B. Wilding, Density De- pletion and Enhanced Fluctuations in Water near Hy- drophobic Solutes: Identifying the Underlying Physics, Phys. Rev. Lett.128, 045501 (2022)

work page 2022

[24] [24]

M. K. Coe, R. Evans, and N. B. Wilding, Understand- ing the physics of hydrophobic solvation, J. Chem. Phys. 158, 034508 (2023)

work page 2023

[25] [25]

Borgis, R

D. Borgis, R. Assaraf, B. Rotenberg, and R. Vuilleumier, Computation of pair distribution functions and three- dimensional densities with a reduced variance principle, Mol. Phys.111, 3486 (2013)

work page 2013

[26] [26]

de las Heras and M

D. de las Heras and M. Schmidt, Better Than Counting: Density Profiles from Force Sampling, Phys. Rev. Lett. 120, 218001 (2018)

work page 2018

[27] [27]

Rotenberg, Use the force! Reduced variance estima- tors for densities, radial distribution functions, and local mobilities in molecular simulations, J

B. Rotenberg, Use the force! Reduced variance estima- tors for densities, radial distribution functions, and local mobilities in molecular simulations, J. Chem. Phys.153, 150902 (2020)

work page 2020

[28] [28]

Renner, M

J. Renner, M. Schmidt, and D. de las Heras, Reduced- variance orientational distribution functions from torque sampling, J. Phys.: Condens. Matter35, 235901 (2023)

work page 2023

[29] [29]

S.M.Tschopp, H.D.Vuijk, A.Sharma,andJ.M.Brader, Mean-field theory of inhomogeneous fluids, Phys. Rev. E 102, 042140 (2020)

work page 2020

[30] [30]

S. M. Tschopp and J. M. Brader, Fundamental measure theory of inhomogeneous two-body correlation functions, Phys. Rev. E103, 042103 (2021)

work page 2021

[31] [31]

S. M. Tschopp, F. Sammüller, S. Hermann, M. Schmidt, and J. M. Brader, Force density functional theory in- and out-of-equilibrium, Phys. Rev. E106, 014115 (2022)

work page 2022

[32] [32]

Sammüller, S

F. Sammüller, S. Hermann, and M. Schmidt, Compara- tive study of force-based classical density functional the- ory, Phys. Rev. E107, 034109 (2023)

work page 2023

[33] [33]

S. N. Jamadagni, R. Godawat, and S. Garde, Hydropho- bicity of Proteins and Interfaces: Insights from Density Fluctuations, Annu. Rev. Chem. Biomol. Eng. 2, 147 (2011)

work page 2011

[34] [34]

Eckert, N

T. Eckert, N. C. X. Stuhlmüller, F. Sammüller, and M. Schmidt, Fluctuation Profiles in Inhomogeneous Flu- ids, Phys. Rev. Lett.125, 268004 (2020)

work page 2020

[35] [35]

Cats and R

P. Cats and R. van Roij, The differential capacitance as a probe for the electric double layer structure and the electrolytebulkcomposition,J.Chem.Phys. 155,104702 (2021)

work page 2021

[36] [36]

E. T. Jaynes, Gibbs vs Boltzmann Entropies, Am. J. Phys. 33, 391 (1965)

work page 1965

[37] [37]

D. C. Wallace, On the role of density fluctuations in the entropy of a fluid, J. Chem. Phys.87, 2282 (1987)

work page 1987

[38] [38]

Schmidt, Statics and dynamics of inhomogeneous liq- uids via the internal-energy functional, Phys

M. Schmidt, Statics and dynamics of inhomogeneous liq- uids via the internal-energy functional, Phys. Rev. E84, 051203 (2011)

work page 2011

[39] [39]

M. J. Mitchell and J. A. McCammon, Free energy dif- ference calculations by thermodynamic integration: Dif- ficulties in obtaining a precise value, J. Comput. Chem. 12, 271 (1991)

work page 1991

[40] [40]

L. S. Ornstein and F. Zernike, Accidental deviations of density and opalescence at the critical point of a single substance, Proc. Am. Acad. Arts Sci.17, 793 (1914)

work page 1914

[41] [41]

Schmidt, Power functional theory for many-body dy- namics, Rev

M. Schmidt, Power functional theory for many-body dy- namics, Rev. Mod. Phys.94, 015007 (2022)

work page 2022

[42] [42]

J. M. Brader and M. Schmidt, Nonequilibrium Ornstein- Zernike relation for Brownian many-body dynamics, J. Chem. Phys.139, 104108 (2013)

work page 2013

[43] [43]

J. M. Brader and M. Schmidt, Dynamic correlations in Brownian many-body systems, J. Chem. Phys.140, 034104 (2014)

work page 2014

[44] [44]

Widom, Some topics in the theory of fluids, J

B. Widom, Some topics in the theory of fluids, J. Chem. Phys. 39, 2808 (1963)

work page 1963

[45] [45]

González, J

A. González, J. A. White, F. L. Román, S. Velasco, and R. Evans, Density functional theory for small systems: Hardspheresinaclosedsphericalcavity,Phys.Rev.Lett. 79, 2466 (1997)

work page 1997

[46] [46]

J. A. White and A. González, The extended variable space approach to density functional theory in the canon- 16 ical ensemble, J. Phys.: Condens. Matter 14, 11907 (2002)

work page 2002

[47] [47]

J. A. White, A. González, F. L. Román, and S. Velasco, Density-functional theory of inhomogeneous fluids in the canonical ensemble, Phys. Rev. Lett.84, 1220 (2000)

work page 2000

[48] [48]

de las Heras and M

D. de las Heras and M. Schmidt, Full canonical infor- mation from grand-potential density-functional theory, Phys. Rev. Lett.113, 238304 (2014)

work page 2014

[49] [49]

de las Heras, T

D. de las Heras, T. Zimmermann, F. Sammüller, S. Her- mann, and M. Schmidt, Perspective: How to overcome dynamical density functional theory, J. Phys.: Condens. Matter 35, 271501 (2023)

work page 2023

[50] [50]

S. P. Kadaoluwa Pathirannahalage, N. Meftahi, A. El- bourne, A. C. G. Weiss, C. F. McConville, A. Padua, D. A. Winkler, M. Costa Gomes, T. L. Greaves, T. C. Le, Q. A. Besford, and A. J. Christofferson, Systematic Comparison of the Structural and Dynamic Properties of Commonly Used Water Models for Molecular Dynamics Simulations, J. Chem. Inf. Model.61, 45...

work page 2021

[51] [51]

J. Lu, Y. Qiu, R. Baron, and V. Molinero, Coarse- Graining of TIP4P/2005, TIP4P-Ew, SPC/E, and TIP3P to Monatomic Anisotropic Water Models Using Relative Entropy Minimization, J. Chem. Theory Com- put. 10, 4104 (2014)

work page 2005

[52] [52]

Molinero and E

V. Molinero and E. B. Moore, Water Modeled As an In- termediateElementbetweenCarbonandSilicon,J.Phys. Chem. B113, 4008 (2009)

work page 2009

[53] [53]

Stierle and J

R. Stierle and J. Gross, Hydrodynamic density functional theory for mixtures from a variational principle and its application to droplet coalescence, J. Chem. Phys.155, 134101 (2021)

work page 2021

[54] [54]

P. M. Rauscher, H. C. Öttinger, and J. J. de Pablo, Nonequilibrium statistical thermodynamics of multicom- ponent interfaces, Proc. Natl. Acad. Sci.119, 24 (2022)

work page 2022

[55] [55]

Hermann and M

S. Hermann and M. Schmidt, Variance of fluctuations from noether invariance, Commun. Phys.5, 276 (2022)

work page 2022

[56] [56]

Hermann and M

S. Hermann and M. Schmidt, Why noether’s theorem ap- plies to statistical mechanics, J. Phys.: Condens. Matter 34, 213001 (2022)

work page 2022

[57] [57]

Hermann and M

S. Hermann and M. Schmidt, Force balance in thermal quantum many-body systems from noether’s theorem, J. Phys. A: Math. Theor.55, 464003 (2022)

work page 2022

[58] [58]

Sammüller, S

F. Sammüller, S. Hermann, D. de las Heras, and M. Schmidt, Noether-constrained correlations in equi- librium liquids (2023), to appear in Phys. Rev. Lett., arXiv:2301.11221 [cond-mat.soft]

work page arXiv 2023

[59] [59]

Prestipino, F

S. Prestipino, F. Saija, and P. V. Giaquinta, Phase di- agram of the Gaussian-core model, Phys. Rev. E 71, 050102 (2005)

work page 2005

[60] [60]

N. B. Wilding, Critical-point and coexistence-curve prop- erties of the Lennard-Jones fluid: A finite-size scaling study, Phys. Rev. E52, 602 (1995)

work page 1995

[61] [61]

Elkamel and R

A. Elkamel and R. D. Noble, Prediction of capillary con- densation in small cylindrical pores using the local den- sity approximation and a full Lennard-Jones 6-12 poten- tial, J. Phys. Chem.95, 10076 (1991)

work page 1991

[62] [62]

U. M. B. Marconi and F. Van Swol, Structure effects and phase-equilibria of Lennard-Jones mixtures in a cylindri- cal pore-a nonlocal density-functional theory, Mol. Phys. 72, 1081 (1991). Appendix A: Functional derivatives In Section IIC, covariance forms of the fluctuation pro- filesχµ(r), χT (r) and χ⋆(r) are derived by explicit eval- uation of the fun...

work page 1991