Statistical Mechanics where Newton's Third Law is Broken

Open Access

Statistical Mechanics where Newton’s Third Law is Broken

A. V. Ivlev, J. Bartnick, M. Heinen, C.-R. Du, V. Nosenko, and H. Löwen

Phys. Rev. X 5, 011035 – Published 26 March 2015

Abstract

There is a variety of situations in which Newton’s third law is violated. Generally, the action-reaction symmetry can be broken for mesoscopic particles, when their effective interactions are mediated by a nonequilibrium environment. Here, we investigate different classes of nonreciprocal interactions relevant to real experimental situations and present their basic statistical mechanics analysis. We show that in mixtures of particles with such interactions, distinct species acquire distinct kinetic temperatures. In certain cases, the nonreciprocal systems are exactly characterized by a pseudo-Hamiltonian; i.e., being intrinsically nonequilibrium, they can nevertheless be described in terms of equilibrium statistical mechanics. Our results have profound implications, in particular, demonstrating the possibility to generate extreme temperature gradients on the particle scale. We verify the principal theoretical predictions in experimental tests performed with two-dimensional binary complex plasmas.

Received 4 March 2014

DOI:https://doi.org/10.1103/PhysRevX.5.011035

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

A. V. Ivlev^1,*, J. Bartnick², M. Heinen^2,3, C.-R. Du⁴, V. Nosenko⁵, and H. Löwen²

¹Max-Planck-Institut für Extraterrestrische Physik, 85741 Garching, Germany
²Institut für Theoretische Physik II, Weiche Materie, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany
³Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA
⁴College of Science, Donghua University, Shanghai 201620, People’s Republic of China
⁵Forschungsgruppe Komplexe Plasmen, Deutsches Zentrum für Luft- und Raumfahrt, 82234 Weßling, Germany

^*ivlev@mpe.mpg.de

Popular Summary

Newton’s third law of action and reaction is one of the cornerstones of classical physics. It holds true not only for microscopic forces but also for effective equilibrium forces between mesoscopic particles. However, this fundamental law can be violated when the effective interaction forces are mediated by a nonequilibrium environment, as often occurs in various soft-matter systems. Prominent examples include colloidal dispersions or complex (dusty) plasmas where the environment—a fluid solvent or a gaseous plasma—can flow with respect to the microparticles or where there is a chemical or thermal imbalance between the particles and molecules of the environment. The symmetry of the interparticle forces in such systems is accordingly broken, so it is a priori unclear whether concepts such as temperature and thermodynamic phases can be used in this case. We present the statistical foundations of systems where the action-reaction symmetry for the interparticle forces is broken.

To describe various classes of nonreciprocal interactions, we consider a generic binary-system model where the action-reaction symmetry is broken for different species. We introduce a nonreciprocity parameter to capture the degree of asymmetry. We rigorously show that in certain cases, the systems with nonreciprocal interactions, being intrinsically nonequilibrium, can nevertheless be described in terms of equilibrium statistical mechanics and can exhibit detailed balance with distinct temperatures for each species. We verify our theoretical predictions in experimental tests with complex plasmas, where two layers of charged microparticles suspended in an argon plasma form a quasi-two-dimensional binary mixture.

Our results lay the groundwork for systematic studies of many-body systems with nonreciprocal interparticle forces and provide a theoretical framework for understanding the self-organization phenomena occurring in weakly damped (e.g., complex plasmas) and strongly damped (e.g., colloidal dispersions) systems with such interactions.

Key Image

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 5, Iss. 1 — January - March 2015

Subject Areas

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Growth of the mean kinetic energy in a 2D binary system (no damping). Particles interact via the nonreciprocal Hertzian forces. The upper panel displays the time dependence of the temperatures $T_{A}$ and $T_{B}$ . The solid lines show the development obtained from the simulations for the areal fraction $ϕ = π r_{0}^{2} n = 0.3$ and different initial temperatures $T_{0}$ . All curves approach the universal asymptotes $\propto t^{2 / 3}$ described by Eqs. (9) and (10). The dotted lines represent the solution of Eq. (7) for $T_{0} ≳ 1$ . The early development at $T_{0} ≪ 1$ is fitted by the explosive solution (13), shown by the dashed lines. The temperatures are normalized by $φ_{0}$ ; time is in units of $\sqrt{m r_{0}^{2} / φ_{0}}$ . The lower panel illustrates the velocity distributions $f_{A, B} (v)$ at $t ≃ 700$ for $T_{0} = 1$ and 10.
Reuse & Permissions
Figure 2
Time dependence of the total kinetic energy (no damping). The development obtained from the simulations for $T_{0} = 1$ and different $ϕ$ (0.1, 0.3, 0.5, 0.7, 0.9, increasing along the arrow) is shown. The inset demonstrates the $\propto n^{2 / 3}$ density scaling of the asymptotic temperature growth. The temperature and time units are the same as in Fig. 1.
Reuse & Permissions
Figure 3
Scheme of the experimental test and the mechanism of nonreciprocal wake-mediated interactions. (a) Sketch showing the experimental setup. Microparticles levitate above a flat horizontal rf electrode. Two sorts of monodisperse particles form two horizontal monolayers at slightly different levitation heights. The layers are observed with the top-view and side-view video cameras, providing complete information about the position of individual particles. (b) Top and side views of a binary mixture forming the two layers. The respective images are obtained by illuminating the particles with thin horizontal and vertical laser sheets. (c) The total force exerted on the upper-layer ( $U$ ) particle from the lower-layer ( $L$ ) particle is the sum of the repulsive force $F_{L U}^{p}$ of direct interparticle interaction and the attractive force $F_{L U}^{w}$ from the wake of the lower particle (and similar for the total force on the lower particle). While the direct forces are reciprocal, $F_{L U}^{p} = - F_{U L}^{p}$ , the wake forces are not, $F_{L U}^{w} \neq - F_{U L}^{w}$ ; since the forces decrease with the distance, we have $| F_{L U}^{w} | < | F_{U L}^{w} |$ and therefore $| F_{L U} | > | F_{U L} |$ . Thus, the total horizontal forces (relevant for the analysis) can be written as $F_{L U} = (1 + Δ) F_{0}$ for the upper layer and $F_{U L} = - (1 - Δ) F_{0}$ for the lower layer.
Reuse & Permissions
Figure 4
Kinetic energy distribution for microparticles in monolayers. Shown are the energy distributions measured for two different values of the rf discharge power $P_{rf}$ , which controls the height difference $H$ between the upper and lower layers. The insets depict the height histograms for particles in the layers; the solid lines are Gaussian fits. (a) $P_{rf} = 20 W$ , $H ≃ 110 μ m$ , and the temperatures of the upper and lower layers are estimated as $T_{U} ≃ 1250 K$ and $T_{L} ≃ 1100 K$ , respectively (from the Maxwellian fit, shown by the dashed lines). (b) $P_{rf} = 10 W$ , $H ≃ 150 μ m$ , and the respective temperatures are $T_{U} ≃ 1350 K$ and $T_{L} ≃ 1000 K$ .
Reuse & Permissions
Figure 5
Sketch illustrating pair collisions in 2D systems. Shown are the variations of the center-of-mass velocity $V$ and the relative velocity $v$ as well as the scattering angle $χ$ , plotted in polar coordinates.
Reuse & Permissions

Physical Review X