Symmetry Properties of the Large-Deviation Function of the Velocity of a Self-Propelled Polar Particle

A geometrically polar granular rod confined in 2D geometry, subjected to a sinusoidal vertical oscillation, undergoes noisy self-propulsion in a direction determined by its polarity. When surrounded by a medium of crystalline spherical beads, it displays substantial negative fluctuations in its velocity. We find that the large-deviation function (LDF) for the normalized velocity is strongly non-Gaussian with a kink at zero velocity, and that the antisymmetric part of the LDF is linear, resembling the fluctuation relation known for entropy production, even when the velocity distribution is clearly non-Gaussian. We extract an analogue of the phase-space contraction rate and find that it compares well with an independent estimate based on the persistence of forward and reverse velocities.

Large deviation function and fluctuation theorem for classical particle transport

We analytically evaluate the large deviation function in a simple model of classical particle transfer between two reservoirs. We illustrate how the asymptotic long-time regime is reached starting from a special propagating initial condition. We show that the steady-state fluctuation theorem holds provided that the distribution of the particle number decays faster than an exponential, implying analyticity of the generating function and a discrete spectrum for its evolution operator.

Quasi-sure existence of Gaussian rough paths and large deviation principles for capacities

We construct a quasi-sure version (in the sense of Malliavin) of geometric rough paths associated with a Gaussian process with long-time memory. As an application we establish a large deviation principle (LDP) for capacities for such Gaussian rough paths. Together with Lyons' universal limit theorem, our results yield immediately the corresponding results for pathwise solutions to stochastic differential equations driven by such Gaussian process in the sense of rough paths. Moreover, our LDP result implies the result of Yoshida on the LDP for capacities over the abstract Wiener space associated with such Gaussian process.

Reply to Comment on ``Towards a large deviation theory for strongly correlated systems''

The computational study commented by Touchette opens the door to a desirable generalization of standard large deviation theory for special, though ubiquitous, correlations. We focus on three interrelated aspects: (i) numerical results strongly suggest that the standard exponential probability law is asymptotically replaced by a power-law dominant term; (ii) a subdominant term appears to reinforce the thermodynamically extensive entropic nature of q-generalized rate function; (iii) the correlations we discussed, correspond to Q -Gaussian distributions, differing from Lévy?s, except in the case of Cauchy?Lorentz distributions. Touchette has agreeably discussed point (i), but, unfortunately, points (ii) and (iii) escaped to his analysis. Claiming the absence of connection with q-exponentials is unjustified.

Guessing Revisited: A Large Deviations Approach

The problem of guessing a random string is revisited. A close relation between guessing and compression is first established. Then it is shown that if the sequence of distributions of the information spectrum satisfies the large deviation property with a certain rate function, then the limiting guessing exponent exists and is a scalar multiple of the Legendre-Fenchel dual of the rate function. Other sufficient conditions related to certain continuity properties of the information spectrum are briefly discussed. This approach highlights the importance of the information spectrum in determining the limiting guessing exponent. All known prior results are then re-derived as example applications of our unifying approach.

Yielding and large deviations in micellar gels: a model

We present a simple model that can be used to account for the rheological behaviour observed in recent experiments on micellar gels. The model combines attachment detachment kinetics with stretching due to shear, and shows well-defined jammed and flowing states. The large-deviation function (LDF) for the coarse-grained velocity becomes increasingly non-quadratic as the applied force F is increased, in a range near the yield threshold. The power fluctuations are found to obey a steady-state fluctuation relation (FR) at small F. However, the FR is violated when F is near the transition from the flowing to the jammed state although the LDF still exists; the antisymmetric part of the LDF is found to be nonlinear in its argument. Our approach suggests that large fluctuations and motion in a direction opposite to an imposed force are likely to occur in a wider class of systems near yielding.

Large deviations for heavy-tailed random elements in convex cones

We prove large deviation results for sums of heavy-tailed random elements in rather general convex cones being semigroups equipped with a rescaling operation by positive real numbers. In difference to previous results for the cone of convex sets, our technique does not use the embedding of cones in linear spaces. Examples include the cone of convex sets with the Minkowski addition, positive half-line with maximum operation and the family of square integrable functions with arithmetic addition and argument rescaling.

Large Deviations and Branching Processes

These lecture notes are devoted to present several uses of Large Deviation asymptotics in Branching Processes.

On the empirical eigenvalue distribution of slotted amplify-and-forward relaying protocol model

In this paper we consider the case of large cooperative communication systems where terminals use the protocol known as slotted amplify-and-forward protocol to aid the source in its transmission. Using the perturbation expansion methods of resolvents and large deviation techniques we obtain an expression for the Stieltjes transform of the asymptotic eigenvalue distribution of a sample covariance random matrix of the type HH† where H is the channel matrix of the transmission model for the transmission protocol we consider. We prove that the resulting expression is similar to the Stieltjes transform in its quadratic equation form for the Marcenko-Pastur distribution.

Aivojen jännitevasteiden rekisteröinti useaan äänen piirteeseen yhdellä mittauskerralla terveillä musiikkileikkikoulua käyvillä kaksivuotiailla

The early detection of hearing deficits is important to a child's development. However, examining small children with behavioural methods is often difficult. Research with ERPs (event-related potentials), recorded with EEG (electroencephalography), does not require attention or action from the child. Especially in children's ERP research, it is essential that the duration of a recording session is not too long. A new, faster optimum paradigm has been developed to record MMN (mismatch negativity), where ERPs to several sound features can be recorded in one recording session. This substantially shortens the time required for the experiment. So far, the new paradigm has been used in adult and school-aged children research. This study examines if MMN, LDN (late discriminative negativity) and P3a components can be recorded in two-year-olds with the new paradigm. The standard stimulus (p=0.50) was an 80 dB harmonic tone consisting of three harmonic frequencies (500 Hz, 1000 Hz and 1500 Hz) with a duration of 200 ms. The loudness deviants (p=0.067) were at a level of +6 dB or -6 dB compared to the standards. The frequency deviants (p=0.112) had a fundamental frequency of 550 or 454.4 Hz (small deviation), 625 or 400 Hz (medium deviation) or 750 or 333.3 Hz (large deviation). The duration deviants (p=0.112) had a duration of 175 ms (small deviation), 150 ms (medium deviation) or 100 ms (large deviation). The direction deviants (p=0.067) were presented from the left or right loudspeaker only. The gap deviant (p=0.067) included a 5-ms silent gap in the middle of the sound. Altogether 17 children participated in the experiment, of whom the data of 12 children was used in the analysis. ERP components were observed for all deviant types. The MMN was significant for duration and gap deviants. The LDN was significant for the large duration deviant and all other deviant types. No significant P3a was observed. These results indicate that the optimum paradigm can be used with two-year-olds. With this paradigm, data on several sound features can be recorded in a shorter time than with the previous paradigms used in ERP research.

Interval Data Classification under Partial Information: A Chance-Constraint Approach

This paper presents a Chance-constraint Programming approach for constructing maximum-margin classifiers which are robust to interval-valued uncertainty in training examples. The methodology ensures that uncertain examples are classified correctly with high probability by employing chance-constraints. The main contribution of the paper is to pose the resultant optimization problem as a Second Order Cone Program by using large deviation inequalities, due to Bernstein. Apart from support and mean of the uncertain examples these Bernstein based relaxations make no further assumptions on the underlying uncertainty. Classifiers built using the proposed approach are less conservative, yield higher margins and hence are expected to generalize better than existing methods. Experimental results on synthetic and real-world datasets show that the proposed classifiers are better equipped to handle interval-valued uncertainty than state-of-the-art.

Robust Formulations for Handling Uncertainty in Kernel Matrices

We study the problem of uncertainty in the entries of the Kernel matrix, arising in SVM formulation. Using Chance Constraint Programming and a novel large deviation inequality we derive a formulation which is robust to such noise. The resulting formulation applies when the noise is Gaussian, or has finite support. The formulation in general is non-convex, but in several cases of interest it reduces to a convex program. The problem of uncertainty in kernel matrix is motivated from the real world problem of classifying proteins when the structures are provided with some uncertainty. The formulation derived here naturally incorporates such uncertainty in a principled manner leading to significant improvements over the state of the art. 1.

Universality and scaling behavior of injected power in elastic turbulence in wormlike micellar gel

We study the statistical properties of spatially averaged global injected power fluctuations for Taylor-Couette flow of a wormlike micellar gel formed by surfactant cetyltrimethylammonium tosylate. At sufficiently high Weissenberg numbers the shear rate, and hence the injected power p(t), at a constant applied stress shows large irregular fluctuations in time. The nature of the probability distribution function (PDF) of p(t) and the power-law decay of its power spectrum are very similar to that observed in recent studies of elastic turbulence for polymer solutions. Remarkably, these non-Gaussian PDFs can be well described by a universal, large deviation functional form given by the generalized Gumbel distribution observed in the context of spatially averaged global measures in diverse classes of highly correlated systems. We show by in situ rheology and polarized light scattering experiments that in the elastic turbulent regime the flow is spatially smooth but random in time, in agreement with a recent hypothesis for elastic turbulence.

Statistical properties of entropy-consuming fluctuations in jammed states of laponite suspensions: Fluctuation relations and generalized Gumbel distribution

We report a universal large deviation behavior of spatially averaged global injected power just before the rejuvenation of the jammed state formed by an aging suspension of laponite clay under an applied stress. The probability distribution function (PDF) of these entropy consuming strongly non-Gaussian fluctuations follow an universal large deviation functional form described by the generalized Gumbel (GG) distribution like many other equilibrium and nonequilibrium systems with high degree of correlations but do not obey the Gallavotti-Cohen steady-state fluctuation relation (SSFR). However, far from the unjamming transition (for smaller applied stresses) SSFR is satisfied for both Gaussian as well as non-Gaussian PDF. The observed slow variation of the mean shear rate with system size supports a recent theoretical prediction for observing GG distribution.