We determine the nonlocal stress autocorrelation tensor in an homogeneous and isotropic system of interacting Brownian particles starting from the Smoluchowski equation of the configurational probability density. In order to relate stresses to particle displacements as appropriate in viscoelastic states, we go beyond the usual hydrodynamic description obtained in the Zwanzig-Mori projection operator formalism by introducing the proper irreducible dynamics following Cichocki and Hess, and Kawasaki. Differently from these authors, we include transverse contributions as well. This recovers the expression for the stress autocorrelation including the elastic terms in solid states as found for Newtonian and Langevin systems, in case that those are evaluated in the overdamped limit. Finally, we argue that the found memory function reduces to the shear and bulk viscosity in the hydrodynamic limit of smooth and slow fluctuations and derive the corresponding hydrodynamic equations.
We have directly observed short-time stress propagation in viscoelastic fluids using two optically trapped particles and a fast interferometric particle-tracking technique. We have done this both by recording correlations in the thermal motion of the particles and by measuring the response of one particle to the actively oscillated second particle. Both methods detect the vortex-like flow patterns associated with stress propagation in fluids. This inertial vortex flow propagates diffusively for simple liquids, while for viscoelastic solutions the pattern spreads super-diffusively, dependent on…
We study many interacting Brownian particles under a tilted periodic potential. We numerically measure the linear response coefficient of the density field by applying a slowly varying potential transversal to the tilted direction. In equilibrium cases, the linear response coefficient is related to the intensity of density fluctuations in a universal manner, which is called a fluctuation-response relation. We then report numerical evidence that this relation holds even in non-equilibrium cases. This result suggests that Einstein's formula on density fluctuations can be extended to driven diffu…
The Ludwig-Soret effect, the migration of a species due to a temperature gradient, has been extensively studied without a complete picture of its cause emerging. Here we investigate the dynamics of DNA and spherical particles sub jected to a thermal gradient using a combination of Brownian dynamics and the lattice Boltzmann method. We observe that the DNA molecules will migrate to colder regions of the channel, an observation also made in the experiments of Duhr, et al[1]. In fact, the thermal diffusion coefficient found agrees quantitatively with the experimental value. We also observe that t…
The Brownian loop soup (BLS) is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity $λ>0$. Recently, we constructed families of operators in the BLS and showed that they transform as conformal primary operators. In this paper we provide an explicit expression for the BLS stress-energy tensor and compute its operator product expansion with other operators. Our results are consistent with the conformal Ward identities and our previous result that the central charge is $c = 2 λ$. In the case of domains with boundary we identify a boundary o…
Tunable shear thickening in active frictional non-Brownian suspensions
School of Engineering, University of Edinburgh（爱丁堡大学工程学院）；Department of Physics, Indian Institute of Science Education and Research Pune（印度科学教育研究所浦那分校物理系）
We use simulations to probe the flow properties of dense two-dimensional magnetorheological fluids. Prior results from both experiments and simulations report that the shear stress $σ$ scales with strain rate $\dot γ$ as $σ\sim \dot γ^{1-Δ}$, with values of the exponent ranging between $2/3 <Δ\le 1$. However it remains unclear what properties of the system select the value of $Δ$, and in particular under what conditions the system displays a yield stress ($Δ= 1$). To address these questions, we perform simulations of a minimalistic model system in which particles interact via long ranged magne…
Motivated from the central role of the mean-square displacement and its second time-derivative -- that is the velocity autocorrelation function $\left\langle v(0)v(t)\right\rangle=\frac{1}{2} \frac{\mathrm{d}^{2} \left\langle Δr^{2} (t)\right\rangle}{\mathrm{d}t^{2}} $ in the description of Brownian motion, we revisit the physical meaning of the first time-derivative of the mean-square displacement of Brownian particles. By employing a rheological analogue for Brownian motion, we show that the time-derivative of the mean-square displacement $\frac{\mathrm{d}\left\langle Δr^{2} (t) \right\rangl…
We consider a Brownian particle in a ``meandering'' periodic potential when the ambient temperature is a periodically or stochastically varying function of time. Though far from equilibrium, the linear response of the particle to an external static force is exactly the same as in the equilibrium case, i.e. for constant temperature. Even more surprising is the non-linear response: the particle slows down and then even starts to move in the direction opposite to the applied force.
We investigate the stress response function of a layer of grains, i.e. the stress profile in response to a localized overload. The shape of the profile is very sensitive to the packing arrangement, and is thus a good signature of the preparation procedure of the layer. This study has been done by the use of molecular dynamics numerical simulations. Here, for a given rain-like preparation, we present the scaling properties of the response function, and in particular the influence of the thickness of the layer, and the importance of the location of the overload and measurement points (at the bou…
We derive the hydrodynamic equations of motion for a fluid of active particles described by under- damped Langevin equations that reduce to the Active-Brownian-Particle model, in the overdamped limit. The contraction into the hydrodynamic description is performed by locally averaging the par- ticle dynamics with the non-equilibrium many-particle probability density, whose formal expression is found in the physically relevant limit of high-friction through a multiple-time-scale analysis. This approach permits to identify the conditions under which self-propulsion can be subsumed into the fluid …
We present a comprehensive study of the linear response of interacting underdamped Brownian particles to simple shear flow. We collect six different routes for computing the response, two of which are based on the symmetry of the considered system and observable with respect to the shear axes. We include the extension of the Green-Kubo relation to underdamped cases, which shows two unexpected additional terms. These six computational methods are applied to investigate the relaxation of the response towards the steady state for different observables, where interesting effects due to interaction…
Considering the paradigmatic driven Brownian motion, we perform extensive numerical analysis on the performance of optimal linear-response processes far from equilibrium. We focus on the overdamped regime where exact optimal processes are known analytically, and most experiments operate. This allows us to compare the optimal processes obtained in linear response and address their relevance to experiments, using realistic parameter values from experiments with optical tweezers. Our results help assess the accuracy of perturbative methods in calculating the irreversible work for cases where an e…
Within the Rayleigh-Helmholtz model of active Brownian particles activity is due to a non-linear velocity dependent force. In the presence of an external trapping potential or a constant force, the steady state of the system breaks detailed balance producing a net entropy. Using molecular dynamics simulations, we obtain the probability distributions of entropy production in these steady states. The distribution functions obey detailed fluctuation theorem for entropy production. Using simulation results, we further show that the steady state response function obeys a modified fluctuation-dissip…
We study the linear response of interacting active Brownian particles in an external potential to simple shear flow. Using a path integral approach, we derive the linear response of any state observable to initiating shear in terms of correlation functions evaluated in the unperturbed system. For systems and observables which are symmetric under exchange of the $x$ and $y$ coordinates, the response formula can be drastically simplified to a form containing only state variables in the corresponding correlation functions (compared to the generic formula containing also time derivatives). In gene…
Basing on main principles of statistical mechanics only, an exact virial expansion for path probability distribution of molecular Brownian particle in a fluid is derived which connects response of the distribution to perturbations of the fluid and statistical correlations of its molecules with Brownian particle. The expansion implies that (i) spatial spread of these correlations is finite, (ii) this is inconsistent with Gaussian distribution involved by the ``molecular chaos'' hypothesis, and (iii) real path distribution possesses power-law long tails. This means that actual Brownian path neve…
Amorphous materials such as metallic, polymeric, and colloidal glasses, exhibit complex preparation-dependent mechanical response to applied shear. We perform numerical simulations to investigate the mechanical response of binary Lennard-Jones glasses undergoing athermal, quasistatic pure shear as a function of the cooling rate $R$ used to prepare them. The ensemble-averaged stress versus strain curve $\langleσ(γ)\rangle$ resembles the spatial average in the large size limit, which appears smooth and displays a putative elastic regime at small strains, a yielding-related peak at intermediate s…
Fluctuation-dissipation relations have received significant attention as a potential method for defining an effective temperature in nonequilibrium systems. The successful development of an effective temperature would be an important step in the application of statistical mechanics principles to systems driven far from equilibrium. Many of the systems of interest are sufficiently dense that they are close to the jamming transition, a point at which interesting correlations develop. Here we study the response function in a driven system of plastic beads as a function of the density in order to …
Athermal systems across a large range of length scales, ranging from foams and granular bead packings to crumpled metallic sheets, exhibit slow stress relaxation when compressed. Experimentally they show a non-monotonic stress response when decompressed somewhat after an initial compression, i.e. under a two-step, Kovacs-like protocol. It turns out that from this response one can tell for how long the system was in a compressed state, suggesting an interpretation as a memory effect. In this work we use a model of an athermal jammed solid, specifically a binary mixture of soft harmonic particle…
We analyze the escape of Brownian particles over potential barriers using the Fokker-Planck equation in a similar way to that of Chandrasekhar (Rev. Modern Phys., 1943), deriving a formula for the particle deposition velocity to a surface as a function of the particle response time. For very small particle response times, the particle deposition velocity reduces to that obtained using a quasi-steady solution of Smoluchowski's equation and for very large particle response times, the deposition velocity corresponds to that based on the transition state method (E. Wigner, Trans. Faraday Soc., 193…
We perform molecular dynamic (MD) simulations of frictional non-thermal particles driven by an externally applied shear stress. After the system jams following a transient flow, we probe its mechanical response in order to clarify whether the resulting solid is 'fragile'. We find the system to respond elastically and isotropically to small perturbations of the shear stress, suggesting absence of fragility. These results are interpreted in terms of the energy landscape of dissipative systems. For the same values of the control parameters, we check the behaviour of the system during a stress cyc…
Exact quantum master equation for a driven Brownian oscillator system is constructed via a Wigner phase-space Gaussian wave packet approach. The interplay between external field and dissipation leads to this system an effective field correction that enhances the polarization. This cooperative property is resulted from an effective bath response to the external field applied on the system. It is important in the low-frequency driving and intermediate bath memory region. We demonstrate this non-Markovian effect on the linear response and nonlinear dynamics and analyze the results together with a…
Brownian motion of an array of harmonically coupled particles subject to a periodic substrate potential and driven by an external bias is investigated. In the linear response limit (small bias), the coupling between particles may enhance the diffusion process, depending on the competition between the harmonic chain and the substrate potential. An analytical formula of the diffusion rate for the single-particle case is also obtained. In the nonlinear response regime, the moving kink may become phase-locked to its radiated phonon waves, hence the mobility of the chain may decrease as one increas…
A steady shear flow can drive supercooled liquids into a non-equilibrium state. Using molecular dynamics simulations under steady shear flow superimposed with oscillatory shear strain for a probe, non-equilibrium mechanical responses are studied for a model supercooled liquid composed of binary soft spheres. We found that even in the strongly sheared situation, the supercooled liquid exhibits surprisingly isotropic responses to oscillating shear strains applied in three different components of the strain tensor. Based on this isotropic feature, we successfully constructed a simple two-mode Max…
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 are exemplified with two prototypical models for the…
We derive the exact expression of the diffusion coefficient of a self-gravitating Brownian gas in two dimensions. Our formula generalizes the usual Einstein relation for a free Brownian motion to the context of two-dimensional gravity. We show the existence of a critical temperature T_{c} at which the diffusion coefficient vanishes. For T<T_{c} the diffusion coefficient is negative and the gas undergoes gravitational collapse. This leads to the formation of a Dirac peak concentrating the whole mass in a finite time. We also stress that the critical temperature T_{c} is different from the colla…
The shear stress autocorrelation function has been studied recently by molecular dynamics simulation using the 1/q^n potential for very large n. The results are analyzed and interpreted here by comparing them to the shear stress response function for hard spheres. It is shown that the hard sphere response function has a singular contribution and that this is reproduced accurately by the simulations for large n. A simple model for the stress autocorrelation function at finite n is proposed, based on the required hard sphere limiting form.
We derive the distribution of particle currents for a system of interacting active Brownian particles in the long time limit using large deviation theory and a weighted many body expansion. We find the distribution is non-Gaussian, except in the limit of passive particles. The non-Gaussian fluctuations can be understood from the effective potential the particles experience when conditioned on a given current. This potential suppresses fluctuations of the particles orientations and surrounding density, aligning particles and reducing their effective drag. From the distribution of currents, we c…
We present a method for computing parameter sensitivities and response coefficients in Brownian dynamics simulations. The method involves tracking auxiliary variables (Malliavin weights) in addition to the usual particle positions, in an unperturbed simulation. The Malliavin weights sample the derivatives of the probability density with respect to the parameters of interest and are also interesting dynamical objects in themselves. Malliavin weight sampling is simple to implement, applies to equilibrium or nonequilibrium, steady state or time-dependent systems, and scales more efficiently than …
