I am fascinated by Soft Matter and its mathematical description, in particular of the fluctuations that occur on small length scales in complex systems. A recent striking example is an experimental system of DNA-coated colloids. These particles are also called nanocaterpillars due to their lepidopteran movement of binding and unbinding their fluctuating legs to a substrate, a process which results in diffusive motion. This behaviour was recently beautifully described based on a mathematical analysis of a one-dimensional model developed by Sophie Marbach et al. [1].
Fluctuations occur naturally in many-body systems in various different ways and there are powerful tools to cope with the seemingly arbitrary consequences that they can induce. These tools include statistical mechanic ensemble averages that endow fluctuating quantities with a well-defined mean and variance. Additionally, exact identities (“sum rules”) allow one to relate fluctuations to further physical quantities. The trusted fluctuation-dissipation theorem is an eminent example thereof.
One prominent way to identify nontrivial identities in physics is based on exploiting Noether’s theorem. In our work [2] we have developed new relations for fluctuations, and we have done so by starting from Noether’s theorem as applied to statistical mechanics.
The theorem itself is of course famous for generating conservation laws from symmetries (or "invariances") of the system. In classical mechanics such a law is e.g. the momentum conservation law which, as it turns out in every respectable undergraduate classical mechanics course, is a direct consequence of a translational invariance of the action functional.
Recently we have shown that the Noether concept transcends to thermal invariances of statistical mechanics functionals [3] (for an introductory account see Ref. [4]). The grand potential, as it is known from elementary statistical mechanics, is in fact a functional of the external potential. One can then show that this thermodynamic potential is invariant with respect to a global homogeneous translation of the system. (Despite being bounded by an external potential, it does not matter at which position our system resides, as long as it is in an isolated box.)
For small amounts of translation one can Taylor expand the functional with respect to the shift parameter and then consider only the linear term. Due to the invariance of the functional with respect to the translation, this term has to vanish. This straightforward argument yields an exact sum rule of vanishing global external force in the system.
However, the translational invariance of the grand potential also holds for finite and even for arbitrarily large shifts. Hence we have gone beyond linear order, as Noether’s theorem applies also to quadratic (and higher-order) terms. In our paper [2] in Communications Physics we determined the sum rule that results from terms in quadratic order from a uniform translation of the grand potential.
The identity states that the covariance of the global external force, which is just the average of the external force correlated with itself, is equal to the mean curvature of the external potential. As the covariance measures the fluctuations of the external force our identity indeed states a fluctuation-relation.
We can bring the fluctuation sum rule into an integral form and this has a nice graphic representation, see Fig. 1(a). (Some readers might prefer to look at the bare formulae as given in the paper instead!) In the Figure the black dots indicate a spatial integration over the entire system volume. The left-hand side denotes the correlation of the external force, given by the negative gradient of the external potential at two different (primed and unprimed) positions multiplied by the important correlation function of the density fluctuations. This term is balanced by the right-hand side, which is the integral over the Hessian of the external potential weighted with the density profile.
Analogously exploiting the translational symmetry of the excess free energy gives a fluctuation identity for the interparticle interactions, see Fig. 1(b) and the paper for the description of the individual terms. Interestingly, the standard variance and fluctuations of the global internal force vanish trivially already due to Newtons third law, which states that force resulting from the interparticle interactions have to vanish for each microstate of the system. Here we can trace this property back to the thermal Noether invariance!
Starting from the translational invariance of the ideal gas free energy functional gives a similar Noether sum rule for the global thermal diffusive forces. This is a nice consistency check, as this relation is just a trivial identity for the Hessian of the logarithmic density and can be proven alternatively by simply integrating that term by parts.
The exact identities that we have developed [2] show that the variance of the global forces equals the mean curvature of the corresponding potential energy function. Such sum rules are essential in liquid state and soft matter theory, they are useful relations for carrying out consistency and validation tests and they can serve as a potential starting point for constructing new theoretical approximations and closure relations. We have recently identified the thermal Noether invariance in quantum system [5]. These systems are still in thermal equilibrium, but we do consider locally resolved force densities.
Hence we are confident that our work could be useful in future investigations of the physics of complex soft matter systems, such as e.g. for the dynamics of crawling nanocaterpillars described above.
References:
[1] S. Marbach, J. A. Zheng, and M. Holmes-Cerfon, The nanocaterpillar’s random walk: diffusion with ligand–receptor contacts, Soft Matter 18, 3130 (2022).
[2] S. Hermann and M. Schmidt, Variance of fluctuations from Noether invariance, Commun. Phys. 5, 276 (2022).
[3] S. Hermann and M. Schmidt, Noether’s theorem in statistical mechanics, Commun. Phys. 4, 176 (2021).
[4] S. Hermann and M. Schmidt, Why Noether's Theorem applies to Statistical Mechanics, J. Phys.: Condens. Matter 34, 213001 (2022).
[5] S. Hermann and M. Schmidt, Force balance in thermal quantum many-body systems from Noether's theorem, J. Phys. A: Math. Theor (to appear in the Special Issue: Claritons and the Asymptotics of ideas: the Physics of Michael Berry).
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