arXiv subjects
Publications and source records attributed to Arnab Saha.
We study the $n$-th arithmetic jet space of the $p$-torsion subgroup attached to a smooth commutative formal group scheme. We show that the $n$-th jet space above fits in the middle of a canonical short exact sequence between a power of the formal scheme of Witt vectors of length $n$ and the $p$-torsion subgroup we started with. This result generalizes a result of Buium on roots of unity.
We perform an extensive analysis of passive as well as active micro-heat engines with different single-particle stochastic models. Using stochastic thermodynamics we calculate thermodynamic work, heat, entropy production and efficiency of passive and active Brownian heat engines analytically as well as numerically and compare them. We run the heat engines with a protocol for which the average thermodynamic quantities are calculated exactly for an arbitrary cycle time. We also discuss about the group of protocols for which exact non-quasistatic calculations can be done, completely in the passive engine case and partially in the active engines. We obtain detailed thermodynamics of non-quasistatic (i.e. powerful) single-particle micro heat engines. The quasistatic (i.e. zero power) limit of the results is obtained by taking long (infinite) cycle time. We also study the distributions of position of the confined particle in both passive and active engines. We compare their characteristics in terms of the parameter that measures the competition between the active persistence in the particle position (due to active noises) and the harmonic confinement. We also calculate excess kurtosis that measures the non-Gaussianity of these distributions. Our analysis shows that efficiency of such thermal machine can be enhanced or reduced depending on the activity present in the model.
We consider binary suspension of rough, circular particles in two dimensions under athermal conditions. The mean density of the system is kept constant. The suspension is subject to a time-independent external drive in response to which one half of the particles are pulled along the field direction whereas the other half is pushed in the opposite direction. Simulating the system with different magnitude of external drive in steady state, we obtain oppositely moving macroscopic lanes only for a moderate range of external drive. Below as well as above the range we obtain states with no lane. Hence we find that no-lane state re-enters along the axis of the external drive in the non-equilibrium phase diagram corresponding to the laning transition, with varying roughness of individual particles and external drive. Inter-particle friction (contact dissipation) due to the roughness of the individual particle is the main player behind the re-entrance of no-lane state at high external drives.
We extend Borger's construction of algebraic jet spaces to allow for an arbitrary prolongation sequence, clarify the relation between Borger's and Buium's jet spaces and compare them in the extended sense. As a result, we strengthen a result of Buium on the relation between Greenberg's transform and the special fiber of jet spaces, including the ramified case.
In this article, we construct differential modular forms for compact Shimura curves over totally real fields bigger than rational of non-zero integral weights that is not classical (of order zero) generalizing the construction of Buium [8].
The topic of microscopic heat engine has undergone intensive research in recent years. Microscopic heat engines can exploit thermal as well as active fluctuations to extract thermodynamic work. We investigate the properties of a microscopic Stirling's engine that uses an active (self-propelling) particle as a working substance, in contact with two thermal baths. It is shown that the presence of activity leads to an enhanced performance of the engine. The efficiency can be improved by increasing the activity strength for all cycle time, including the non-quasistatic regime. We verify that the analytical results agree very well with our simulations. The variation of efficiency with the temperature difference between the two thermal baths has also been explored. The optimum region of operation of the engine has been deduced, by using its efficient power as a quantifier. Finally, a simple model is provided that emulates the behaviour of a flywheel driven by this engine.
From synthetic active devices such as self-propelling Janus colloids to micro-organisms like bacteria, micro-algae, living cells in tissues, active fluctuations are ubiquitous. Thermodynamics of small systems involving thermal as well as active fluctuations are of immense importance. They can be employed to extract thermodynamic work. Here we propose a simple model system that can produce thermodynamic work exploiting active fluctuations. We consider a Brownian particle, trapped by an externally controlled harmonic confinement which time-periodically contracts and expands by modulating its spring constant e.g an optical tweezer. The system produces work by being alternately connected to two baths one passive and other active, modeled as exponentially correlated noise which breaks the fluctuation dissipation theorem. The average efficiency of the system is calculated exactly in quasistatic limit. Nonquasistatic regime is explored by numerics. Comparing with its passive counterpart, we also show that the active micro heat engine can be more efficient depending on the chosen parameter space. We also believe that our model can be realized experimentally with the help of bacterial baths.
Using Buium's theory of arithmetic differential characters, we construct a filtered $F$-isocrystal ${\bf H}(A)_K$ associated to an abelian scheme $A$ over a $p$-adically complete discrete valuation ring with perfect residue field. As a filtered vector space, ${\bf H}(A)_K$ admits a natural map to the usual de Rham cohomology of $A$, but the Frobenius operator comes from arithmetic differential theory and is not the same as the usual crystalline one. When $A$ is an elliptic curve, we show that ${\bf H}(A)_K$ has a natural integral model ${\bf H}(A)$, which implies an integral refinement of a result of Buium's on arithmetic differential characters. The weak admissibility of ${\bf H}(A)_K$ depends on the invertibility of an arithmetic-differential modular parameter. Thus the Fontaine functor associates to suitably generic $A$ a local Galois representation of an apparently new kind.
We introduce differential characters of Drinfeld modules. These are function-field analogues of Buium's p-adic differential characters of elliptic curves and of Manin's differential characters of elliptic curves in differential algebra, both of which have had notable Diophantine applications. We determine the structure of the group of differential characters. This shows the existence of a family of interesting differential modular functions on the moduli of Drinfeld modules. It also leads to a canonical $F$-crystal equipped with a map to the de Rham cohomology of the Drinfeld module. This $F$-crystal is of a differential-algebraic nature, and the relation to the classical cohomological realizations is presently not clear.
We consider collective dynamics of self-propelling particles in two dimensions. They can align themselves according to the direction of propulsion of their neighbours, together with a random perturbation (i.e. rotational fluctuation). They are also interacting with each other by a soft, isotropic, excluded-volume interaction. Particles are confined in a circular trap. The steepness of the trap is tuneable. Their average packing fraction and strength of rotational fluctuation are low. When the trap is steep, particles flock along its boundary. They form polar cluster that spreads over the boundary. The cluster has no spatial or structural order. We show, when the steepness is decreased beyond a threshold value, the clusters become round, compact and eventually spatial order (hexagonal) emerges in addition to the pre-established polar order within them. We investigate kinetics of such ordering. We find that while rotating around the centre of the trap along its circular boundary, the clusters need to roll around their centre of mass, to be spatially ordered. We also discuss the stability of the order when the trap is suddenly switched off.
We investigate the effect of cooperative interactions in an ensemble of microorganisms, modelled as self-propelled disk-like and rod-like particles, in a three-dimensional turbulent flow to show flocking as an emergent phenomenon. Building on the work by Choudhary, et al. [Europhys. Lett. 112, 24005 (2015)] for two-dimensional systems, and combining ideas from active matter and turbulent transport, we show that non-trivial correlations between the flow and individual dynamics are essential for the microorganisms to flock in, for example, a turbulent three-dimensional, marine environment. Our results may have implications especially in the modelling of artificial microswimmers in a hostile environment.
Thermodynamics of nanoscale devices is an active area of research. Despite their noisy surrounding they often produce mechanical work (e.g. micro-heat engines), display rectified Brownian motion (e.g. molecular motors). This invokes research in terms of experimentally quantifiable thermodynamic efficiencies. Here, a Brownian particle is driven by a harmonic confinement with time-periodic contraction and expansion. The system produces work by being alternately (time-periodically) connected to baths with different dissipations. We analyze the system theoretically using stochastic thermodynamics. Averages of thermodynamic quantities like work, heat, efficiency, entropy are found analytically for long cycle times. Simulations are also performed in various cycle-times. They show excellent agreement with analytical calculations in the long cycle time limit. Distributions of work, efficiency, and large deviation function for efficiency are studied using simulations. We believe that the experimental realization of our model is possible.
We consider flocking of self-propelling agents in two dimensions, each of which communicates with its neighbors within a limited vision cone. Also, the communication occurs with some delay. The communication among the agents are modeled by Vicsek rules. In this study we explore the effect of non-reciprocal interaction among the agents, induced by their vision cone, together with the delayed interactions on the dynamical pattern formation within the flock. We find that under these two influences and without any position based attractive interactions or confining boundaries, the agents can spontaneously condense into drops. Though the agents are in motion within the drop, the drop as whole is virtually pinned in space. We also find that this novel state of the flock has a well defined order stabilized by the noise present in the system.
In differential algebra, a proper scheme $X$ defined over an algebraically closed field $K$ with a derivation $\partial$ on it descends to the field of constants $K^\partial$ if $X$ itself lifts the derivation $\partial$. This is a result by A. Buium. Now in the arithmetic case, the notion of a derivation is replaced by the notion of a $π$-derivation $δ$ or equivalently in the flat case, a lift of Frobenius $ϕ$. We will show an analogous result in the arithmetic case of equal characteristic. We show our results using the arithmetic analogue of Taylor expansion using Witt vectors.
In this article, given a scheme $X$ we show the existence of canonical lifts of Frobenius maps in an inverse system of schemes obtained from the fiber product of the canonical prolongation sequence of arithmetic jet spaces $J^*X$ and a prolongation sequence $S^*$ over the scheme $X$. As a consequence, for any smooth group scheme $E$, if $N^n$ denote the kernel of the canonical projection map of the $n$-th jet space $J^nE \rightarrow E$, then the inverse system $\{N^n\}_n$ is a prolongation sequence.
Micro-to-nano scale thermal devices that operate under large thermal fluctuations, are an active field of research where instead the average values, the full distributions of thermodynamic quantities are important. Here we study a model of stochastic heat engine consisting of a harmonically trapped Brownian particle driven by the time-periodic strength of the confinement, within two thermal baths of different temperatures. The particle follows two isotherms correspond to two baths and connected by two micro-adiabates. The microadiabaticity is implemented by conserving the phase space volume of the particle along the adiabatic paths. Here we show that it can operate as an engine or as a heater under microadiabaticity, depending on the parameter space. We also compute the distribution of stochastic efficiency and its averages for different cycle times of the engine.
Thermodynamics of nanoscale devices is an active area of research. Despite their noisy surround- ing they often produce mechanical work (e.g. micro-heat engines) or display rectified Brownian motion (e.g. molecular motors). This invokes the research in terms of experimentally quantifiable thermodynamic efficiencies. To enhance the efficiency of such devices, close-loop control is an useful technique. Here a single Brownian particle is driven by a harmonic confinement with time-periodic contraction and expansion, together with a velocity feedback that acts on the particle only when the trap contracts. Due to this feedback we are able to extract thermodynamic work out of the system having single heat bath without violating the Second Law of Thermodynamics. We analyse the system using stochastic thermodynamics.
Over the last century Bohr van Leuween theorem attracted the notice of physicists. The theorem states about the absence of magnetization in classical systems in thermal equilibrium. In this paper, we discuss about fluctuations of magnetic moment in classical systems. In recent years this topic has been investigated intensively and it is not free from controversy. We a have considered a system consisting of a single particle moving in a plane. A magnetic field is applied perpendicular to the plane. The system is in contact with a thermal bath. We have considered three cases: (a) particle moving in a homogeneous medium, (b) particle moving in a medium with space dependent friction and (c) particle moving in a medium with space dependent temperature. For all the three cases average magnetic moment and fluctuations in magnetic moment has been calculated. Average magnetic moment saturates to a finite value in case of free particle but goes to zero when the particle is confined by a 2-D harmonic potential. Fluctuations in magnetic moment shows universal features in the presence of arbitrary friction inhomogeneity. For this case the system reaches equilibrium asymptotically. In case of space dependent temperature profile, the stationary distribution is non-Gibbsian and fluctuations deviate from universal value for the bounded system only.