scholarly journals Work fluctuations in the active Ornstein–Uhlenbeck particle model

2021 ◽  
Vol 2021 (12) ◽  
pp. 123202
Author(s):  
Massimiliano Semeraro ◽  
Antonio Suma ◽  
Isabella Petrelli ◽  
Francesco Cagnetta ◽  
Giuseppe Gonnella
Keyword(s):  
Numerical Solutions ◽  
Free Particle ◽  
Bath Temperature ◽  
Rate Function ◽  
Potential Interaction ◽  
Two Dimensions ◽  
Particle Model ◽  
Confining Potential ◽  
Fenchel Transform ◽  
Work Distribution

Abstract We study the large deviations of the power injected by the active force for an active Ornstein–Uhlenbeck particle (AOUP), free or in a confining potential. For the free-particle case, we compute the rate function analytically in d-dimensions from a saddle-point expansion, and numerically in two dimensions by (a) direct sampling of the active work in numerical solutions of the AOUP equations and (b) Legendre–Fenchel transform of the scaled cumulant generating function obtained via a cloning algorithm. The rate function presents asymptotically linear branches on both sides and it is independent of the system’s dimensionality, apart from a multiplicative factor. For the confining potential case, we focus on two-dimensional systems and obtain the rate function numerically using both methods (a) and (b). We find a different scenario for harmonic and anharmonic potentials: in the former case, the phenomenology of fluctuations is analogous to that of a free particle, but the rate function might be non-analytic; in the latter case the rate functions are analytic, but fluctuations are realised by entirely different means, which rely strongly on the particle-potential interaction. Finally, we check the validity of a fluctuation relation for the active work distribution. In the free-particle case, the relation is satisfied with a slope proportional to the bath temperature. The same slope is found for the harmonic potential, regardless of activity, and for an anharmonic potential with low activity. In the anharmonic case with high activity, instead, we find a different slope which is equal to an effective temperature obtained from the fluctuation–dissipation theorem.

scholarly journals Wave-induced stress and breaking of sea ice in a coupled hydrodynamic–discrete-element wave–ice model

2017 ◽  
Author(s):  
Agnieszka Herman
Keyword(s):  
Sea Ice ◽  
Wave Attenuation ◽  
Wave Model ◽  
Discrete Element ◽  
Two Dimensions ◽  
Particle Model ◽  
Time Step ◽  
Bonded Particle Model ◽  
Induced Stress ◽  
Wave Induced

Abstract. In this paper, a coupled sea ice–wave model is developed and used to analyze the variability of wave-induced stress and breaking in sea ice. The sea ice module is a discrete-element bonded-particle model, in which ice is represented as cuboid "grains" floating on the water surface that can be connected to their neighbors by elastic "joints". The joints may break if instantaneous stresses acting on them exceed their strength. The wave part is based on an open-source version of the Non-Hydrostatic WAVE model (NHWAVE). The two parts are coupled with proper boundary conditions for pressure and velocity, exchanged at every time step. In the present version, the model operates in two dimensions (one vertical and one horizontal) and is suitable for simulating compact ice in which heave and pitch motion dominates over surge. In a series of simulations with varying sea ice properties and incoming wavelength it is shown that wave-induced stress reaches maximum values at a certain distance from the ice edge. The value of maximum stress depends on both ice properties and characteristics of incoming waves, but, crucially for ice breaking, the location at which the maximum occurs does not change with the incoming wavelength. Consequently, both regular and random (Jonswap spectrum) waves break the ice into floes with almost identical sizes. The width of the zone of broken ice depends on ice strength and wave attenuation rates in the ice.

Numerical simulation of two-dimensional compressible convection

1975 ◽  
Vol 70 (4) ◽  
pp. 689-703 ◽  
Author(s):  
Eric Graham
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Fluid Velocity ◽  
Critical Rayleigh Number ◽  
Finite Amplitude ◽  
Two Dimensions ◽  
Onset Of Convection ◽  
Local Sound ◽  
Constant Viscosity ◽  
Convection Problem

A procedure for obtaining numerical solutions to the equations describing thermal convection in a compressible fluid is outlined. The method is applied to the case of a perfect gas with constant viscosity and thermal conductivity. The fluid is considered to be confined in a rectangular region by fixed slippery boundaries and motions are restricted to two dimensions. The upper and lower boundaries are maintained at fixed temperatures and the side boundaries are thermally insulating. The resulting convection problem can be characterized by six dimension-less parameters. The onset of convection has been studied both by obtaining solutions to the nonlinear equations in the neighbourhood of the critical Rayleigh number Rc and by solving the linear stability problem. Solutions have been obtained for values of the Rayleigh number up to 100Rc and for pressure variations of a factor of 300 within the fluid. In some cases the fluid velocity is comparable to the local sound speed. The Nusselt number increases with decreasing Prandtl number for moderate values of the depth parameter. Steady finite amplitude solutions have been found in all the cases considered. As the horizontal dimension A of the rectangle is increased, the length of time needed to reach a steady state also increases. For large values of A the solution consists of a number of rolls. Even for small values of A, no solutions have been found where one roll is vertically above another.

Path integrals in polar coordinates

1969 ◽  
Vol 313 (1514) ◽  
pp. 445-452 ◽  
Keyword(s):  
Path Integrals ◽  
Free Particle ◽  
Two Dimensions ◽  
Polar Coordinates ◽  
Hamiltonian Approach

The problem of summation over Feynman histories in polar coordinates is discussed. Ordinary rules of calculus are used in conjunction with a Hamiltonian approach to the summation over histories. The procedure is checked by evaluating the propagator for a free particle in two dimensions.

scholarly journals Mechanical torque promotes bipolarity of the mitotic spindle through multi-centrosomal clustering

2021 ◽  
Author(s):  
Christopher E Miles ◽  
Jie Zhu ◽  
Alex Mogilner
Keyword(s):  
Mitotic Spindle ◽  
Numerical Solutions ◽  
Interaction Model ◽  
Particle Model ◽  
Cellular Organization ◽  
Discrete Particle Model ◽  
A Cell ◽  
Intracellular Forces ◽  
And Function ◽  
Dynamic Microtubule

Intracellular forces shape cellular organization and function. One example is the mitotic spindle, a cellular machine consisting of multiple chromosomes and centrosomes which interact via dynamic microtubule filaments and motor proteins, resulting in complicated spatially dependent forces. For a cell to divide properly, is important for the spindle to be bipolar, with chromosomes at the center and multiple centrosomes clustered into two 'poles' at opposite sides of the chromosomes. Experimental observations show that in unhealthy cells, the spindle can take on a variety of patterns. What forces drive each of these patterns? It is known that attraction between centrosomes is key to bipolarity, but what the prevents the centrosomes from collapsing into a monopolar configuration? Here, we explore the hypothesis that torque rotating chromosome arms into orientations perpendicular to the centrosome-centromere vector promotes spindle bipolarity. To test this hypothesis, we construct a pairwise-interaction model of the spindle. On a continuum version of the model, an integro-PDE system, we perform linear stability analysis and construct numerical solutions which display a variety of spatial patterns. We also simulate a discrete particle model resulting in a phase diagram that confirms that the spindle bipolarity emerges most robustly with torque. Altogether, our results suggest that rotational forces may play an important role in dictating spindle patterning.

scholarly journals An age-structured continuum model for myxobacteria

2018 ◽  
Vol 28 (09) ◽  
pp. 1737-1770 ◽  
Author(s):  
Pierre Degond ◽  
Angelika Manhart ◽  
Hui Yu
Keyword(s):  
Molecular Mechanisms ◽  
Ad Hoc ◽  
Order Approximation ◽  
Continuous Model ◽  
Two Dimensions ◽  
Macroscopic Model ◽  
Particle Model ◽  
Diffusion Term ◽  
Age Structured ◽  
Age Structured Model

Myxobacteria are social bacteria, that can glide in two dimensions and form counter-propagating, interacting waves. Here, we present a novel age-structured, continuous macroscopic model for the movement of myxobacteria. The derivation is based on microscopic interaction rules that can be formulated as a particle-based model and set within the Self-Organized Hydrodynamics (SOH) framework. The strength of this combined approach is that microscopic knowledge or data can be incorporated easily into the particle model, whilst the continuous model allows for easy numerical analysis of the different effects. However, we found that the derived macroscopic model lacks a diffusion term in the density equations, which is necessary to control the number of waves, indicating that a higher order approximation during the derivation is crucial. Upon ad hoc addition of the diffusion term, we found very good agreement between the age-structured model and the biology. In particular, we analyzed the influence of a refractory (insensitivity) period following a reversal of movement. Our analysis reveals that the refractory period is not necessary for wave formation, but essential to wave synchronization, indicating separate molecular mechanisms.

An exact treatment of collisional effects on temporal plasma wave echoes: free-particle model

1970 ◽  
Vol 4 (4) ◽  
pp. 843-847 ◽  
Author(s):  
Y. Furutani ◽  
J. Coste
Keyword(s):  
Exact Solution ◽  
Plasma Wave ◽  
Free Particle ◽  
Particle Model ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field

In the case of a free-particle plasma with no self-consistent field and with collisions described by the LenardBernstein operator we give an exact solution of the temporal echo problem.

Nonlinear drift phase-space structures

1992 ◽  
Vol 47 (3) ◽  
pp. 401-429
Author(s):  
Sharadini Rath ◽  
P. K. Kaw
Keyword(s):  
Phase Space ◽  
Numerical Solutions ◽  
Physical Space ◽  
Nonlinear Effects ◽  
Two Dimensions ◽  
Space Structures ◽  
Stationary States ◽  
Relative Importance ◽  
Poisson System ◽  
The Magnetic Field

The collisionless Vlasov–Poisson system in the drift approximation is examined for the existence of maximum-entropy nonlinear coherent solutions in the steady state. Two major nonlinear effects are taken into account. The first is the velocity-space trapping of particles, leading to closed trajectories in phase space. The second is the physical-space trapping of particles, leading to closed trajectories in the plane perpendicular to the magnetic field. The regions of validity of these nonlinearities are discussed and their relative importance demonstrated. Numerical solutions of the equations describing the nonlinear stationary states in one and two dimensions are presented.

scholarly journals Partial regularisation of the incompressible 𝜇(I)-rheology for granular flow

2017 ◽  
Vol 828 ◽  
pp. 5-32 ◽  
Author(s):  
T. Barker ◽  
J. M. N. T. Gray
Keyword(s):  
High Speed ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Functional Dependence ◽  
Granular Flows ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Major Step ◽  
Ill Posed ◽  
Well Posed

In recent years considerable progress has been made in the continuum modelling of granular flows, in particular the $\unicode[STIX]{x1D707}(I)$-rheology, which links the local viscosity in a flow to the strain rate and pressure through the non-dimensional inertial number $I$. This formulation greatly benefits from its similarity to the incompressible Navier–Stokes equations as it allows many existing numerical methods to be used. Unfortunately, this system of equations is ill posed when the inertial number is too high or too low. The consequence of ill posedness is that the growth rate of small perturbations tends to infinity in the high wavenumber limit. Due to this, numerical solutions are grid dependent and cannot be taken as being physically realistic. In this paper changes to the functional form of the $\unicode[STIX]{x1D707}(I)$ curve are considered, in order to maximise the range of well-posed inertial numbers, while preserving the overall structure of the equations. It is found that when the inertial number is low there exist curves for which the equations are guaranteed to be well posed. However when the inertial number is very large the equations are found to be ill posed regardless of the functional dependence of $\unicode[STIX]{x1D707}$ on $I$. A new $\unicode[STIX]{x1D707}(I)$ curve, which is inspired by the analysis of the governing equations and by experimental data, is proposed here. In order to test this regularised rheology, transient granular flows on inclined planes are studied. It is found that simulations of flows, which show signs of ill posedness with unregularised models, are numerically stable and match key experimental observations when the regularised model is used. This paper details two-dimensional transient computations of decelerating flows where the inertial number tends to zero, high-speed flows that have large inertial numbers, and flows which develop into granular rollwaves. This is the first time that granular rollwaves have been simulated in two dimensions, which represents a major step towards the simulation of other complex granular flows.

scholarly journals Traveling wave solution of the Hele–Shaw model of tumor growth with nutrient

2014 ◽  
Vol 24 (13) ◽  
pp. 2601-2626 ◽  
Author(s):  
Benoît Perthame ◽  
Min Tang ◽  
Nicolas Vauchelet
Keyword(s):  
Free Boundary ◽  
Tumor Growth ◽  
Traveling Wave ◽  
Wave Solution ◽  
Numerical Solutions ◽  
Two Dimensions ◽  
Necrotic Core ◽  
Traveling Wave Solution ◽  
Cancer Evolution

Several mathematical models of tumor growth are now commonly used to explain medical observations and predict cancer evolution based on images. These models incorporate mechanical laws for tissue compression combined with rules for nutrients availability which can differ depending on the situation under consideration, in vivo or in vitro. Numerical solutions exhibit, as expected from medical observations, a proliferative rim and a necrotic core. However, their precise profiles are rather complex, both in one and two dimensions. We study a simple free boundary model formed of a Hele–Shaw equation for the cell number density coupled to a diffusion equation for a nutrient. We can prove that a traveling wave solution exists with a healthy region separated from the progressing tumor by a sharp front (the free boundary) while the transition to the necrotic core is smoother. Remarkable is the pressure distribution which vanishes at the boundary of the proliferative rim with a vanishing derivative at the transition point to the necrotic core.

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