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

Eckert, Tobias; Stuhlmüller, Nico C. X.; Sammüller, Florian; Schmidt, Matthias

doi:10.1088/1361-648x/ace50c

Cited by 3 publications

(2 citation statements)

References 61 publications

(145 reference statements)

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“…Hence the Noether correlation theory does not interfere with approaches that specifically target on particle number effects, such as the local compressibility introduced by Evans and coworkers [84][85][86]. The Noether approach captures genuinely the mechanical fluctuations, as opposed to local chemical (particle number) [53,[84][85][86][87] and thermal (energy) fluctuations [53,86,87]. We lastly point to the recent hyperforce generalization [31] that arises from thermal Noether invariance and that applies to arbitrary observables.…”

Section: Discussionmentioning

confidence: 88%

“…That equation ( 27) holds can be seen by writing out explicitly the thermal average on the left hand side and functionally differentiating all dependencies on the external potential. As an aside, Eckert et al [53] have recently pointed out that for an arbitrary phase space function Â, which is taken to be independent of V ext (r), one has δ⟨ Â⟩/δV ext (r) = −βcov( Â, ρ(r)), which upon choosing Â = F(r) generates the first term on the right hand side of equation ( 27);…”

Section: Density-force Noether Identitymentioning

confidence: 99%

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Noether invariance theory for the equilibrium force structure of soft matter

Hermann,

Sammüller,

Schmidt

2024

J. Phys. A: Math. Theor.

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We give details and derivations for the Noether invariance theory that characterizes the spatial equilibrium structure of inhomogeneous classical many-body systems, as recently proposed and investigated for bulk systems [F. Sammüller et al., Phys. Rev. Lett. 130, 268203 (2023)]. Thereby an intrinsic thermal symmetry against a local shifting transformation on phase space is exploited on the basis of the Noether theorem for invariant variations. We consider the consequences of the shifting that emerge at second order in the displacement field that parameterizes the transformation. In a natural way the standard two-body density distribution is generated. Its second spatial derivative (Hessian) is thereby balanced by two further and different two-body correlation functions, which respectively introduce thermally averaged force correlations and force gradients in a systematic and microscopically sharp way into the framework. Separate exact self and distinct sum rules hold expressing this balance. We exemplify the validity of the theory on the basis of computer simulations for the Lennard-Jones gas, liquid, and crystal, the Weeks-Chandler-Andersen fluid, monatomic Molinero-Moore water at ambient conditions, a three-body-interacting colloidal gel former, the Yukawa and soft-sphere dipolar fluids, and for isotropic and nematic phases of Gay-Berne particles. We describe explicitly the derivation of the sum rules based on Noether's theorem and also give more elementary proofs based on partial phase space integration following Yvon's theorem.

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Section: Discussionmentioning

confidence: 88%

Section: Density-force Noether Identitymentioning

confidence: 99%

Noether invariance theory for the equilibrium force structure of soft matter

Hermann,

Sammüller,

Schmidt

2024

J. Phys. A: Math. Theor.

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Hyperforce balance via thermal Noether invariance of any observable

Robitschko,

Sammüller,

Schmidt

et al. 2024

Commun Phys

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Noether invariance in statistical mechanics provides fundamental connections between the symmetries of a physical system and its conservation laws and sum rules. The latter are exact identities that involve statistically averaged forces and force correlations and they are derived from statistical mechanical functionals. However, the implications for more general observables and order parameters are unclear. Here, we demonstrate that thermally averaged classical phase space functions are associated with exact hyperforce sum rules that follow from translational Noether invariance. Both global and locally resolved identities hold and they relate the mean gradient of a phase-space function to its negative mean product with the total force. Similar to Hirschfelder’s hypervirial theorem, the hyperforce sum rules apply to arbitrary observables in equilibrium. Exact hierarchies of higher-order sum rules follow iteratively. As applications we investigate via computer simulations the emerging one-body force fluctuation profiles in confined liquids. These local correlators quantify spatially inhomogeneous self-organization and their measurement allows for the development of stringent convergence tests and enhanced sampling schemes in complex systems.

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What is the best simulation approach for measuring local density fluctuations near solvo-/hydrophobes?

Wilding,

Evans,

Turci

2024

The Journal of Chemical Physics

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Measurements of local density fluctuations are crucial to characterizing the interfacial properties of equilibrium fluids. A specific case that has been well-explored involves the heightened compressibility of water near hydrophobic entities. Commonly, a spatial profile of local fluctuation strength is constructed from the measurements of the mean and variance of solvent particle number fluctuations in a set of contiguous subvolumes of the system adjacent to the solvo-/hydrophobe. An alternative measure proposed by Evans and Stewart [J. Phys.: Condens. Matter 27, 194111 (2015)] defines a local compressibility profile in terms of the chemical potential derivative of the spatial number density profile. Using Grand canonical Monte Carlo simulation, we compare and contrast the efficacy of these two approaches for a Lennard-Jones solvent at spherical and planar solvophobic interfaces and SPC/E water at a hydrophobic spherical solute. Our principal findings are as follows: (i) the local compressibility profile χ(r) of Evans and Stewart is considerably more sensitive to variations in the strength of local density fluctuations than the spatial fluctuation profile F(r) and can resolve much more detailed structure; and (ii) while the local compressibility profile is essentially independent of the choice of spatial discretization used to construct the profile, the spatial fluctuation profile exhibits a strong systematic dependence on the size of the subvolumes on which the profile is defined. We clarify the origin and nature of this finite-size effect.

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

Cited by 3 publications

References 61 publications

Noether invariance theory for the equilibrium force structure of soft matter

Noether invariance theory for the equilibrium force structure of soft matter

Hyperforce balance via thermal Noether invariance of any observable

What is the best simulation approach for measuring local density fluctuations near solvo-/hydrophobes?

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