Heterogeneous diffusion with stochastic resetting

2022 ◽  
Author(s):  
Trifce Sandev ◽  
Viktor Domazetoski ◽  
Ljupco Kocarev ◽  
Ralf Metzler ◽  
Alexei Chechkin
Keyword(s):  
Steady State ◽  
Diffusion Process ◽  
Inner Core ◽  
Numerical Solutions ◽  
Diffusion Length ◽  
Large Deviation ◽  
Nonequilibrium Steady State ◽  
Length Scale ◽  
Normal Diffusion ◽  
Long Time

Abstract We study a heterogeneous diffusion process with position-dependent diffusion coefficient and Poissonian stochastic resetting. We find exact results for the mean squared displacement and the probability density function. The nonequilibrium steady state reached in the long time limit is studied. We also analyze the transition to the non-equilibrium steady state by finding the large deviation function. We found that similarly to the case of the normal diffusion process where the diffusion length grows like $t^{1⁄2}$ while the length scale ξ(t) of the inner core region of the nonequilibrium steady state grows linearly with time t, in the heterogeneous diffusion process with diffusion length increasing like $t^{p⁄2}$ the length scale ξ(t) grows like $t^{p}$. The obtained results are verified by numerical solutions of the corresponding Langevin equation.

scholarly journals Mathematical Modelling of the Effects of Mitotic Inhibitors on Avascular Tumour Growth

1999 ◽  
Vol 1 (4) ◽  
pp. 287-311 ◽  
Author(s):  
J. P. Ward ◽  
J. R. King
Keyword(s):  
Steady State ◽  
Tumour Growth ◽  
Numerical Solutions ◽  
Mitotic Rate ◽  
Travelling Wave ◽  
Living Cells ◽  
Cellular Material ◽  
Raw Material ◽  
Long Time ◽  
Mitotic Inhibitors

In this paper we build on the mathematical model of Ward and King (1998) to study the effects of high molecular mass mitotic inhibitors released at cell death. The model assumes a continuum of living cells which, depending on the concentration of a generic nutrient, generate movement (described by a velocity field) due to the changes in volumes caused by cell birth and death. The necrotic material is assumed to consist of two diffusible materials: 1) basic cellular material which is used by living cells as raw material for mitosis; 2) a generic non-utilisable material which may inhibit mitosis. Numerical solutions of the resulting system of partial differential equations show all the main features of tumour growth and heterogeneity. Material 2) is found to act in an inhibitive fashion in two ways: i) directly, by reducing the mitotic rate and ii) indirectly, by occupying space, thereby reducing the availability of the basic cellular material. For large time the solutions to the model tend either to a steady-state, reflecting growth saturation, or to a travelling wave, indicating continual linear growth. The steady-state and travelling wave limits of the model are derived and studied, the regions of existence of these two types of long-time solution being explored in parameter space using numerical methods.

scholarly journals Modelling the Effect of Cell Shedding on Avascular Tumour Growth

2000 ◽  
Vol 2 (3) ◽  
pp. 155-174 ◽  
Author(s):  
J. P. Ward ◽  
J. R. King
Keyword(s):  
Steady State ◽  
Tumour Growth ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Tumour Size ◽  
Travelling Wave ◽  
Live Cells ◽  
Long Time ◽  
Cell Shedding

Earlier mathematical models of the authors which describe avascular tumour growth are extended to incorporate the process of cell shedding, a feature known to affect the growth of multicell spheroids. A continuum of live cells is assumed within which, depending on the concentration of a generic nutrient, movement (described by a velocity field) occurs due to volume changes caused by cell birth and death. The necrotic material is assumed to contain a mixture of basic cellular material (assumed necessary for creating new cells) and a non-utilisable material which may inhibit mitosis. The rate of cell shedding is taken to be proportional to the mitotic rate, with constant of proportionality θ. Numerical solutions of the resulting system of partial differential equations indicate that, depending on θ and the initial conditions, the solution may either tend to the trivial state in finite time (by which we mean complete death of the tumour), or to one of two non-trivial states, namely a steady-state (indicating growth saturation) or a travelling wave (indicating continual linear growth). These long time outcomes are explored by deriving the travelling wave and steady-state limits of the model. Numerical solutions demonstrate that there are two branches of solutions, which we have termed the ′Major′ and ′Minor′ branches, consisting of both travelling waves and steady-states. The behaviour of the solutions along each branch is discussed, with those of the Major branch expected to be stable. Beyond some critical θ,where the Major and Minor branches merge, the spheroid ultimately vanishes whatever the initial tumour size due to the effects of cell shedding being too strong for it to survive. The regions of existence of the two long time outcomes are investigated in parameter space, cell shedding being shown to expand significantly the parameter ranges within which growth saturation occurs.

Effects of slide-to-roll ratio and varying velocity on the lubrication performance of grease at low speed

2021 ◽  
pp. 135065012199336
Author(s):  
Yiming Han ◽  
Jing Wang ◽  
Xuyang Jin ◽  
Shanshan Wang ◽  
Rui Zhang
Keyword(s):  
Steady State ◽  
Critical Speed ◽  
Industrial Applications ◽  
Transient Effects ◽  
Disintegration Time ◽  
Low Speed ◽  
Existence Time ◽  
Pure Rolling ◽  
Lubrication Performance ◽  
Long Time

Under steady-state pure rolling conditions with low speed, the thickener fiber agglomerations can be maintained for a long time, generating a beneficial thicker film thickness. However, in industrial applications, motions with sliding or transient effects are very common for gears, rolling-element bearings or even chain drives, evaluation of the grease performance under such conditions is vital for determining the lubrication mechanism and designing new greases. In this project, optical interferometry experiments were carried out on a ball-disk test rig to study the disintegration time of the grease thickener agglomerations with the increase of the slide-to-roll ratio under steady-state and reciprocation motions. Under steady-state conditions, the thickener fiber agglomeration can exist for a while and the time becomes shorter with the increase of the slide-to-roll ratio above the critical speed. Below the critical speed, the thickener fiber can exist in the contact in the form of a quite thick film for a very long time under pure rolling conditions but that time is decreased with the increase of the slide-to-roll ratio. The introduction of the transient effect can further reduce the existence time of the thickener.

scholarly journals Perturbation of eigenvalues due to gaps in two-dimensional boundaries

2006 ◽  
Vol 463 (2079) ◽  
pp. 759-786 ◽  
Author(s):  
Anthony M.J Davis ◽  
Stefan G Llewellyn Smith
Keyword(s):  
Eigenvalue Problems ◽  
Numerical Solutions ◽  
Neumann Boundary ◽  
Diffusion Problem ◽  
Length Scale ◽  
Two Dimensional ◽  
Constructive Approach ◽  
Term Outcome ◽  
Long Term Outcome

Motivated by problems involving diffusion through small gaps, we revisit two-dimensional eigenvalue problems with localized perturbations to Neumann boundary conditions. We recover the known result that the gravest eigenvalue is O (|ln  ϵ | −1 ), where ϵ is the ratio of the size of the hole to the length-scale of the domain, and provide a simple and constructive approach for summing the inverse logarithm terms and obtaining further corrections. Comparisons with numerical solutions obtained for special geometries, both for the Dirichlet ‘patch problem’ where the perturbation to the boundary consists of a different boundary condition and for the gap problem, confirm that this approach is a simple way of obtaining an accurate value for the gravest eigenvalue and hence the long-term outcome of the underlying diffusion problem.

Super-Fine Structure in the Critical Flow-Rate of Critical Flow Venturi Nozzles

2002 ◽  
Author(s):  
Masahiro Ishibashi
Keyword(s):  
Fine Structure ◽  
Steady State ◽  
Discharge Coefficient ◽  
Constant Pressure ◽  
Pressure Ratio ◽  
Critical Flow ◽  
Time Intervals ◽  
Critical Flow Rate ◽  
A Value ◽  
Long Time

It is shown that critical flow Venturi nozzles need time intervals, i.e., more than five hours, to achieve steady state conditions. During these intervals, the discharge coefficient varies gradually to reach a value inherent to the pressure ratio applied. When a nozzle is suddenly put in the critical condition, its discharge coefficient is trapped at a certain value then afterwards approaches gradually to the inherent value. Primary calibrations are considered to have measured the trapped discharge coefficient, whereas nozzles in applications, where a constant pressure ratio is applied for a long time, have a discharge coefficient inherent to the pressure ratio; inherent and trapped coefficients can differ by 0.03–0.04%.

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