Probability Density of the Fractional Langevin Equation with Reflecting Walls

We investigate anomalous diffusion processes governed by the fractional Langevin equation and confined to a finite or semi-infinite interval by reflecting potential barriers. As the random and damping forces in the fractional Langevin equation fulfill the appropriate fluctuation-dissipation relation, the probability density on a finite interval converges for long times towards the expected uniform distribution prescribed by thermal equilibrium. In contrast, on a semi-infinite interval with a reflecting wall at the origin, the probability density shows pronounced deviations from the Gaussian behavior observed for normal diffusion. If the correlations of the random force are persistent (positive), particles accumulate at the reflecting wall while antipersistent (negative) correlations lead to a depletion of particles near the wall. We compare and contrast these results with the strong accumulation and depletion effects recently observed for nonthermal fractional Brownian motion with reflecting walls, and we discuss broader implications

Fractional Brownian Motion in a Finite Interval: Correlations Effect Depletion or Accretion Zones of Particles Near Boundaries

Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) ⟨X2(t)⟩ ≃ tα with 1 \u3c α \u3c 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 \u3c α \u3c 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers

Fractional Brownian Motion in a Finite Interval: Correlations Effect Depletion or Accretion Zones of Particles Near Boundaries

Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) ⟨X2(t)⟩ ≃ tα with 1 \u3c α \u3c 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 \u3c α \u3c 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers

Prospective Prediction of Posttraumatic Stress Disorder Symptoms Using Fear Potentiated Auditory Startle Responses

Background: Posttraumatic stress disorder (PTSD) has been most consistently associated with exaggerated physiologic reactivity to startling sounds when such sounds occur in threatening contexts. There is conflicting evidence about whether startle hyperreactivity is a preexisting vulnerability factor for PTSD or an acquired result of posttrauma neural sensitization. Until now, there have been no prospective studies of physiologic reactivity to startling sounds in threatening contexts as predictors of PTSD symptoms. Methods: One hundred and thirty-eight police academy cadets without current psychopathology were exposed to repeated 106-dB startling sounds under increasing (low, medium, or high) threat of mild electric shock while their eye-blink electromyogram, skin conductance, heart rate, and subjective fear responses were recorded. Measures of response habituation were also calculated. Following 1 year of exposure to police-related trauma, these participants were assessed for PTSD symptom severity. Results: After accounting for other baseline variables that were predictive of PTSD symptom severity (age and general psychiatric distress), more severe PTSD symptoms were prospectively and independently predicted by the following startle measures: greater subjective fear under low threat, greater skin conductance under high threat, and slower skin conductance habituation. Conclusions: These results imply that hypersensitivity to contextual threat (indexed by greater fear under low threat), elevated sympathetic nervous system reactivity to explicit threat (indexed by larger responses under high threat), and failure to adapt to repeated aversive stimuli (evidenced by slower habituation) are all unique preexisting vulnerability factors for greater PTSD symptom severity following traumatic stress exposure. These measures may eventually prove useful for preventing PTSD

Reflected fractional Brownian motion in one and higher dimensions

Fractional Brownian motion (FBM), a non-Markovian self-similar Gaussian stochastic process with long-ranged correlations, represents a widely applied, paradigmatic mathematical model of anomalous diffusion. We report the results of large-scale computer simulations of FBM in one, two, and three dimensions in the presence of reflecting boundaries that confine the motion to finite regions in space. Generalizing earlier results for finite and semi-infinite one-dimensional intervals, we observe that the interplay between the long-time correlations of FBM and the reflecting boundaries leads to striking deviations of the stationary probability density from the uniform density found for normal diffusion. Particles accumulate at the boundaries for superdiffusive FBM while their density is depleted at the boundaries for subdiffusion. Specifically, the probability density $P$ develops a power-law singularity, $P\sim r^\kappa$, as function of the distance $r$ from the wall. We determine the exponent $\kappa$ as function of the dimensionality, the confining geometry, and the anomalous diffusion exponent $\alpha$ of the FBM. We also discuss implications of our results, including an application to modeling serotonergic fiber density patterns in vertebrate brains.Comment: 14 pages, 20 figures included, final version as publishe

Gender and PTSD: What Can We Learn from Female Police Officers?

Studies of civilians typically find that female gender is a risk factor for posttraumatic stress disorder (PTSD). Police and military studies often find no gender differences in PTSD. We compared 157 female police officers and 124 female civilians on several variables including trauma exposure, peritraumatic emotional distress, current somatization, and cumulative PTSD symptoms. We found that despite greater exposure to assaultive violence in the officer group, female civilians reported significantly more severe PTSD symptoms. Elevated PTSD symptoms in female civilians were explained by significantly more intense peritraumatic emotional distress among female civilians. We also found that female officers showed a stronger direct relationship between peritraumatic emotional distress and current somatization. Our findings suggest that apparent gender differences in PTSD may result from differences in peritraumatic emotionality, which influence subsequent PTSD and somatization symptoms. Emotionality may be more important than biological sex in understanding gender differences in PTSD

A note on confined diffusion

The random motion of a Brownian particle confined in some finite domain is considered. Quite generally, the relevant statistical properties involve infinite series, whose coefficients are related to the eigenvalues of the diffusion operator. Unfortunately, the latter depend on space dimensionality and on the particular shape of the domain, and an analytical expression is in most circumstances not available. In this article, it is shown that the series may in some circumstances sum up exactly. Explicit calculations are performed for 2D diffusion restricted to a circular domain and 3D diffusion inside a sphere. In both cases, the short-time behaviour of the mean square displacement is obtained.Comment: 10 pages; Eq. (2) correcte