Effective temperatures of hot Brownian motion

Linking fluctuation and dissipation in spatially extended out-of-equilibrium systems

For systems in equilibrium at a temperature T, thermal noise and energy damping are related to T through the fluctuation-dissipation theorem (FDT). We study here an extension of the FDT to an out-of-equilibrium steady state: a microcantilever subject to a constant heat flux. The resulting thermal profile in this spatially extended system interplays with the local energy dissipation field to prescribe the amplitude of mechanical fluctuations. Using three samples with different damping profiles (localized or distributed), we probe this approach and experimentally demonstrate the link between fluctuations and dissipation. The thermal noise can therefore be predicted a priori from the measurement of the dissipation as a function of the maximum temperature of the micro-oscillator.

Active Refrigerators Powered by Inertia

We present the operational principle for a refrigerator that uses inertial effects in active Brownian particles to locally reduce their (kinetic) temperature by 2 orders of magnitude below the environmental temperature. This principle exploits the peculiar but so-far unknown shape of the phase diagram of inertial active Brownian particles to initiate motility-induced phase separation in the targeted cooling regime only. Remarkably, active refrigerators operate without requiring isolating walls opening the route toward using them to systematically absorb and trap, e.g., toxic substances from the environment.

Motion of a self-propelled particle with rotational inertia

Overdamped active Brownian motion of self-propelled particles in a liquid has been fairly well studied. However, there are a variety of situations in which the overdamped approximation is not justified, for instance, when self-propelled particles move in a low-viscosity medium or when their rotational diffusivity is enhanced by internal active processes or external control. Examples of various origins include biofilaments driven by molecular motors, living and artificial microflyers and interfacial surfers, field-controlled and superfluid microswimmers, vibration-driven granular particles and autonomous mini-robots with sensorial delays, etc. All of them extend active Brownian motion to the underdamped case, i.e., to active Langevin motion, which takes into account inertia. Despite a rich experimental background, there is a gap in the theory in the field where rotational inertia significantly affects the random walk of active particles on all time scales. In particular, although the well-known models of active Brownian and Ornstein-Uhlenbeck particles include a memory effect of the direction of motion, they are not applicable in the underdamped case, because the rotational inertia, which they do not account for, can partially prevent "memory loss" with increasing rotational diffusion. We describe the two-dimensional motion of a self-propelled particle with both translational and rotational inertia and velocity fluctuations. The proposed generalized analytical equations for the mean kinetic energy, mean-square displacement and noise-averaged trajectory of the self-propelled particle are confirmed by numerical simulations in a wide range of self-propulsion velocities, moments of inertia, rotational diffusivities, medium viscosities and observation times.

Tethered tracer in a mixture of hot and cold Brownian particles: can activity pacify fluctuations?

We study how an interacting mixture of components with differing levels of activity can affect the fluctuations of an embedded object such as a tracer. In particular, we consider a simple model of a tracer that is harmonically bound within a mixture of hot and cold Brownian particles, which, like a mixture of active and passive particles, can phase separate. By measuring the fluctuations of the tracer, we find that this collective behavior gives rise to an effective temperature for the tracer. Additionally, we find that there is an increased tendency for cold particles to accumulate on the surface of the tracer due to the hot particles, potentially dampening its fluctuations and decreasing its effective temperature. These results suggest that the phase separation of a mixture of hot/cold or active/passive particles may have strong effects on the fluctuations of an embedded object. We discuss potential implications of these results for experiments on fluctuations of nuclear envelope affected by the activity in the chromatin.

Fluctuation-dissipation relations far from equilibrium: a case study

Fluctuation-dissipation relations or "theorems" (FDTs) are fundamental for statistical physics and can be rigorously derived for equilibrium systems. Their applicability to non-equilibrium systems is, however, debated. Here, we simulate an active microrheology experiment, in which a spherical colloid is pulled with a constant external force through a fluid, creating near-equilibrium and far-from-equilibrium systems. We characterize the structural and dynamical properties of these systems, and reconstruct an effective generalized Langevin equation (GLE) for the colloid dynamics. Specifically, we test the validity of two FDTs: The first FDT relates the non-equilibrium response of a system to equilibrium correlation functions, and the second FDT relates the memory friction kernel in the GLE to the stochastic force. We find that the validity of the first FDT depends strongly on the strength of the external driving: it is fulfilled close to equilibrium and breaks down far from it. In contrast, we observe that the second FDT is always fulfilled. We provide a mathematical argument why this generally holds for memory kernels reconstructed from a deterministic Volterra equation for correlation functions, even for non-stationary non-equilibrium systems. Motivated by the Mori-Zwanzig formalism, we therefore suggest to impose an orthogonality constraint on the stochastic force, which is in fact equivalent to the validity of this Volterra equation. Such GLEs automatically satisfy the second FDT and are unique, which is desirable when using GLEs for coarse-grained modeling.

Introducing Memory in Coarse-Grained Molecular Simulations

Preserving the correct dynamics at the coarse-grained (CG) level is a pressing problem in the development of systematic CG models in soft matter simulation. Starting from the seminal idea of simple time-scale mapping, there have been many efforts over the years toward establishing a meticulous connection between the CG and fine-grained (FG) dynamics based on fundamental statistical mechanics approaches. One of the most successful attempts in this context has been the development of CG models based on the Mori-Zwanzig (MZ) theory, where the resulting equation of motion has the form of a generalized Langevin equation (GLE) and closely preserves the underlying FG dynamics. In this Review, we describe some of the recent studies in this regard. We focus on the construction and simulation of dynamically consistent systematic CG models based on the GLE, both in the simple Markovian limit and the non-Markovian case. Some recent studies of physical effects of memory are also discussed. The Review is aimed at summarizing recent developments in the field while highlighting the major challenges and possible future directions.

Active Brownian particle in homogeneous media of different viscosities: numerical simulations

Self-propelled colloids, active polymers and membranes, driven (vibrated) granular layers and hybrid synthetic-biological systems are striking examples of systems containing synthetic active Brownian particles. Such particles autonomously convert the available energy of the environment into their own directed mechanical motion. In most studies the self-propelled Brownian particles move in overdamped media. Recently, experiments with Janus particles in a low-pressure plasma have appeared. A distinctive feature of such a medium is an extremely low viscosity at which the inertial effects play a significant role, resulting in underdamped Brownian motion. At present, there is a lack of statistical theory describing the underdamped Brownian motion of self-propelled particles at all time scales. This paper presents the numerical simulation results of active Brownian motion in homogeneous media of different viscosities. The calculations are performed using a mathematical model of a self-propelled Brownian sphere with translational and rotational inertia. The time-dependent mean square displacement and mean linear displacement (the noise-averaged trajectory) of the particle are investigated as a function of medium viscosity, self-propulsion velocity and moment of inertia. Our simulation reveals that the dynamics of a self-propelled spherical particle significantly depends on two independent dimensionless parameters of the particle: the ratio of the self-propulsion velocity to the characteristic thermal velocity and the ratio of the friction coefficient to the rotational diffusion coefficient. The obtained statistical characteristics of active Brownian motion are compared with the known theoretical models in a wide range of medium viscosities. We propose simple corrections to the basic theory of overdamped active Brownian motion, which allow one to calculate the effective diffusion coefficient and the persistence length of a self-propelled Brownian particle in a medium with any dynamic viscosity. The results obtained are discussed in relation to active particles in a colloidal plasma and superfluid helium.

Three-body problem for Langevin dynamics with different temperatures

A mixture of Brownian particles at different temperatures has been a useful model for studying the out-of-equilibrium properties of systems made up of microscopic components with differing levels of activity. This model was previously studied analytically for two-particle interactions in the dilute limit, yielding a Boltzmann-like two-particle distribution with an effective temperature. Like the Newtonian two- and three-body problems, we ask here whether the two-particle results can be extended to three-particle interactions to get the three-particle distributions. By considering the special solvable case of pairwise quadratic interactions, we show that, unlike the two-particle distribution, the three-particle distribution cannot in general be Boltzmann-like with an effective temperature. We instead find that the steady-state distribution of any two particles in a triplet depends on the properties of and interactions with the third particle, leading to some unexpected behaviors not present in equilibrium.

Plasmonic Heating of Nanostructures

The absorption of light by plasmonic nanostructures and their associated temperature increase are exquisitely sensitive to the shape and composition of the structure and to the wavelength of light. Therefore, much effort is put into synthesizing novel nanostructures for optimized interaction with the incident light. The successful synthesis and characterization of high quality and biocompatible plasmonic colloidal nanoparticles has fostered numerous and expanding applications, especially in biomedical contexts, where such particles are highly promising for general drug delivery and for tomorrow's cancer treatment. We review the thermoplasmonic properties of the most commonly used plasmonic nanoparticles, including solid or composite metallic nanoparticles of various dimensions and geometries. Common methods for synthesizing plasmonic particles are presented with the overall goal of providing the reader with a guide for designing or choosing nanostructures with optimal thermoplasmonic properties for a given application. Finally, the biocompatibility and biological tolerance of structures are critically discussed along with novel applications of plasmonic nanoparticles in the life sciences.

Orientational dynamics of a heated Janus particle

Using large scale molecular dynamics simulations, we study the orientational dynamics of a heated Janus particle which exhibits self-propulsion. The asymmetry in the microscopic interaction of the colloid with the solvent is implemented by choosing different wetting parameters for the two halves of the sphere. This choice leads to a different microscopic Kapitza resistance across the solid-fluid boundary of the two halves of the sphere, and consequently a gradient in temperature is created across the poles of the sphere. It is this self-created temperature gradient which leads to a self-propulsion along the direction of the symmetry axis. In this article, we look at the orientational dynamics of such a system, as well as the subsequent enhancement of the translational diffusivity of the heated Janus colloid at late times. The orientational correlation of the symmetry axis is measured from the simulation and provides a direct access to the rotational diffusion constant. The heating leads to an increase in the rotational diffusivity of the colloid. We quantify this increase in rotational diffusion D _r against the temperature difference δT ≡ T(R, 0) - T(R, π) across the poles of the Janus sphere as well as the average surface temperature difference ΔT ≡ T(R) - T(∞) from the ambient fluid. Since the rotational diffusion is determined by the complete flow field in the solvent, we illustrate that comparing D _r against δT is misleading and is better quantified when compared against ΔT. The later quantification results in a data collapse for different choices of the microscopic interaction. The average propulsion velocity is also measured for different choices of the wetting parameter. The directionality of self-propulsion changes depending on the microscopic interaction. We show that whenever the attractive interaction of the colloid with the solvent is switched off, the phoretic mobility changes sign. Furthermore, the propulsion velocity is zero for heating below a certain threshold value. This is also corroborated by the probability distribution of the angle between the displacement vector Δr(t) ≡ r(t) - r(0) and the symmetry axis. Finally, we combine the measured propulsion velocity and the rotational diffusion time τ _r = 1/2D _r to estimate the enhancement in the long time diffusion coefficient of the particle.

The effective temperature for the thermal fluctuations in hot Brownian motion

We revisit the effective parameter description of hot Brownian motion-a scenario where a colloidal particle is kept at an elevated temperature than the ambient fluid. Due to the time scale separation between heat diffusion and particle motion, a stationary halo of hot fluid is carried along with the particle resulting in a spatially varying comoving temperature and viscosity profile. The resultant Brownian motion in the overdamped limit can be well described by a Langevin equation with effective parameters such as effective temperature T_HBM and friction coefficient ζ_HBM that quantifies the thermal fluctuations and the diffusivity of the particle. These parameters can exactly be calculated using the framework of fluctuating hydrodynamics and require the knowledge of the complete flow field and the temperature field around the particle. Additionally, it was also observed that configurational and kinetic degrees of freedom admit to different effective temperatures, T_HBM^x and T_HBM^v, respectively, with the former predicted accurately from fluctuating hydrodynamics. A more rigorous calculation by Falasco et al. [Phys. Rev. E 90, 032131-10 (2014)] extends the overdamped description to a generalized Langevin equation where the effective temperature becomes frequency dependent and consequently, for any temperature measurement from a Brownian trajectory requires the knowledge of this frequency dependence. We use this framework to expand on the earlier work and look at the first order correction to the limiting values in the hydrodynamic limit and the kinetic limit. We use the linearized Stokes equation and a constant viscosity approximation to calculate the dissipation function in the fluid. The effective temperature is calculated from the weighted average of the temperature field with the dissipation function. Further, we provide a closed form analytical result for effective temperature in the small as well as high frequency limit. Since hot Brownian motion can be used to probe the local environment in complex systems, we have also calculated the effective diffusivity of the particle in the small frequency limit. To look into the kinetic temperature, the velocity autocorrelation function is computed from the generalized Langevin equation and the Wiener-Khinchine theorem and numerically integrated to evaluate T_HBM^v as a function of the ratio of particle density and fluid density ρ_P/ρ₀. The two limiting cases of ρ_P/ρ₀ → 0 and ρ_P/ρ₀ → ∞ is also discussed.

Levitated Nanoparticles for Microscopic Thermodynamics-A Review

Levitated Nanoparticles have received much attention for their potential to perform quantum mechanical experiments even at room temperature. However, even in the regime where the particle dynamics are purely classical, there is a lot of interesting physics that can be explored. Here we review the application of levitated nanoparticles as a new experimental platform to explore stochastic thermodynamics in small systems.

Interacting Brownian dynamics in a nonequilibrium particle bath

We set up a mesoscopic theory for interacting Brownian particles embedded in a nonequilibrium environment, starting from the microscopic interacting many-body theory. Using nonequilibrium linear-response theory, we characterize the effective dynamical interactions on the mesoscopic scale and the statistics of the nonequilibrium environmental noise, arising upon integrating out the fast degrees of freedom. As hallmarks of nonequilibrium, the breakdown of the fluctuation-dissipation and action-reaction relations for Brownian degrees of freedom is exemplified with two prototypical models for the environment, namely active Brownian particles and stirred colloids.

Exact symmetries in the velocity fluctuations of a hot Brownian swimmer

Symmetries constrain dynamics. We test this fundamental physical principle, experimentally and by molecular dynamics simulations, for a hot Janus swimmer operating far from thermal equilibrium. Our results establish scalar and vectorial steady-state fluctuation theorems and a thermodynamic uncertainty relation that link the fluctuating particle current to its entropy production at an effective temperature. A Markovian minimal model elucidates the underlying nonequilibrium physics.

Nonisothermal fluctuating hydrodynamics and Brownian motion

The classical theory of Brownian dynamics follows from coarse graining the underlying linearized fluctuating hydrodynamics of the solvent. We extend this procedure to globally nonisothermal conditions, requiring only a local thermal equilibration of the solvent. Starting from the conservation laws, we establish the stochastic equations of motion for the fluid momentum fluctuations in the presence of a suspended Brownian particle. These are then contracted to the nonisothermal generalized Langevin description of the suspended particle alone, for which the coupling to stochastic temperature fluctuations is found to be negligible under typical experimental conditions.

Gold Nanorod Rotary Motors Driven by Resonant Light Scattering

Efficient and robust artificial nanomotors could provide a variety of exciting possibilities for applications in physics, biology and chemistry, including nanoelectromechanical systems, biochemical sensing, and drug delivery. However, the application of current man-made nanomotors is limited by their sophisticated fabrication techniques, low mechanical output power and severe environmental requirements, making their performance far below that of natural biomotors. Here we show that single-crystal gold nanorods can be rotated extremely fast in aqueous solutions through optical torques dominated by plasmonic resonant scattering of circularly polarized laser light with power as low as a few mW. The nanorods are trapped in 2D against a glass surface, and their rotational dynamics is highly dependent on their surface plasmon resonance properties. They can be kept continuously rotating for hours with limited photothermal side effects and they can be applied for detection of molecular binding with high sensitivity. Because of their biocompatibility, mechanical and thermal stability, and record rotation speeds reaching up to 42 kHz (2.5 million revolutions per minute), these rotary nanomotors could advance technologies to meet a wide range of future nanomechanical and biomedical needs in fields such as nanorobotics, nanosurgery, DNA manipulation and nano/microfluidic flow control.

Stochastic heating of a single Brownian particle by charge fluctuations in a radio-frequency produced plasma sheath

The Brownian motion of a single particle in the plasma sheath is studied to separate the effect of stochastic heating by charge fluctuations from heating by collective effects. By measuring the particle velocities in the ballistic regime and by carefully determining the particle mass from the Epstein drag it is shown that for a pressure of 10 Pa, which is typical of many experiments, the proper kinetic temperature of the Brownian particle remains close to the gas temperature and rises only slightly with particle size. This weak effect is confirmed by a detailed model for charging and charge fluctuations in the sheath. A substantial temperature rise is found for decreasing pressure, which approximately shows the expected scaling with p(-2). The system under study is an example for non-equilibrium Brownian motion under the influence of white noise without corresponding dissipation.