arXiv subjects

Arnab Saha

Publications and source records attributed to Arnab Saha.

At least 19 recordsLinked to original sources

arXiv2026

Delta theory of Anderson Modules II: Hodge-Pink structure

In this article, using the theory of $\delta$-geometry, we construct a canonical $z$-isocrystal $(\mathbf{H}_\delta(E), \mathfrak{f}^*)$ admitting a Hodge-Pink structure for any abelian Anderson module $E$. The Hodge-Pink structure on $\mathbf{H}_\delta(E)$ induces a natural filtration $(\mathbf{H}_\delta(E) \supset \mathbf{X}_{\mathrm{prim}}(E) \supset \{0\})$. The elements of $\mathbf{X}_{\mathrm{prim}}(E)$ are represented by primitive delta characters associated to $E$. We establish a natural morphism from $\mathbf{H}_\delta(E)$ to the associated de Rham cohomology module $\mathbf{H}^*_{\mathrm{dR}}(E)$, which is strictly compatible with the aforementioned filtration and the classical Hodge filtration $(\mathbf{H}^{*}_{\mathrm{dR}}(E)\supset {\mathrm{Lie}(E)^{*}}\supset \{0\})$ on $\mathbf{H}^*_{\mathrm{dR}}(E)$. Moreover, we show that the map induces an isomorphism between $\mathbf{X}_{\mathrm{prim}}(E)$ and $\mathrm{Lie}(E)^*$. Hence our isomorphism provides an interesting interpretation of the invariant differentials of $E$ as primitive delta characters of $E$. Furthermore, when $E$ is a Drinfeld module, we show that the constructed $z$-isocrystal $\mathbf{H}_\delta(E)$ is weakly admissible. Consequently, the positive equal characteristic analogue of the Fontaine functor associates a crystalline $z$-adic Galois representation to the $\delta$-geometric object $\mathbf{H}_\delta(E)$. In the case, when $E$ is the Carlitz module, we show that the Galois representation associated to $\mathbf{H}_\delta(E)$ is indeed the usual one coming from the Tate module.

math.NT↗

arXiv2026

Non-reciprocity drives a Brownian dimer out of equilibrium

We consider the minimal model of a two dimensional Brownian dimer consisting of two overdamped monomers, trapped in an isotropic harmonic potential and mutually coupled by a non-reciprocal harmonic spring that violates Newton's action-reaction principle. We have shown that the non-reciprocal interaction alone can drive the system far from equilibrium, in the absence of any external time dependent drive and being in contact with a single thermal bath. The exact steady state probability distribution and current are explicitly calculated for the zero-rest-length limit of the spring, which eventually maps our model to another non-equilibrium phenomenon, called Brownian gyration. For a spring with finite rest length, these quantities are calculated numerically.

cond-mat.stat-mech↗

arXiv2026

Brownian gyration of an inertial ellipsoid

Recent studies on Brownian gyration (BG) have focused primarily on spherically symmetric particles under overdamped conditions. To explore BG in the underdamped regime with a spherically asymmetric particle, we investigate the inertial dynamics of a microscopic ellipsoid in a dissipative medium. The particle is confined in a spherically asymmetric trap and simultaneously coupled to two distinct thermal reservoirs. This configuration drives the system into a nonequilibrium steady state (NESS) characterised by BG, which is quantified by the mean and fluctuation of the particle's specific angular momentum. Using inertial Langevin dynamics, we systematically analyze how this microscopic gyration depends not merely on the trap asymmetry and temperature difference, but also on the particle's intrinsic physical properties like shape and axial orientation, besides inertia. Our study uncovers fundamental differences between the gyration of spherical and non-spherical particles in overdamped as well as underdamped conditions, at microscopic scales. These findings provide key insights for optimizing Brownian gyration across a broader landscape of experimentally tuneable parameters.

cond-mat.stat-mech↗

arXiv2026

From Global Flocking to Local Clustering: Interplay between Velocity Alignment and Visual Perception of Active Particles

Collective behavior in biological systems was first captured by the Vicsek model, in which particles align their velocities in the average direction of neighbors, leading to coherent motion and showing an order-disorder transition. However, in many complex environments, the interactions are non-reciprocal, lacking an action-reaction symmetry. Using framework of the Vicsek model, we implement non-reciprocity by restricting interactions to neighbors located inside a finite vision cone, for a particle by limiting its set of interacting neighbors which fall within a vision-cone, providing a minimal description for cognitive perception. Using detailed numerical simulations, we explore the clustering and flocking behavior due to competition between noise and limited visual perception in the presence of alignment interaction. For low noise, with reduction in the vision angle the system shows transition from a global coherent motion to locally ordered small-sized clusters. This behavior is confirmed through the steady-state distributions of velocity components and their fluctuation relative to the global mean. This is also characterized using a polar order-parameter and a two-point velocity correlation function. Interestingly, at small vision angles, particles exhibit strong short-range correlations within clusters even in the absence of any global coherence. Time-evolution of the related correlation functions illustrate the pathways towards the emergence of such structures. The time dependence of the average cluster size and the length-scale calculated from the two-point velocity correlation show scaling behavior and indicate that the emergence of density field clustering is a consequence of the velocity-field coherence. Any kind of ordering and clustering disappear in the limit of high noise and low vision-angle regime.

cond-mat.soft↗

arXiv2026

Kernels of Arithmetic Jet Spaces and Frobenius Morphism

For any $π$-formal group scheme $G$, the Frobenius morphism between arithmetic jet spaces restricts to generalized kernels of the projection map. Using the functorial properties of such kernels of arithmetic jet spaces, we show that this morphism is indeed induced by a natural ring map between shifted $π$-typical Witt vectors. In the special case when $G = \hat{\mathbb{G}}_a$, the arithmetic jet space, as well as the generalized kernels are affine $π$-formal planes with Witt vector addition as the group law. In that case the above morphism is the multiplication by $π$ map on Witt vector schemes. In fact, the system of arithmetic jet spaces and generalized kernels of any $π$-formal group scheme $G$ along with their maps and identitites satisfied among them are a generalization of the case of the Witt vector scheme with the system of maps such as the Frobenius, Verschiebung and multiplication by $π$.

math.AG↗

arXiv2025

Miniature work-to-work converter engine powered by motor protein

Designing a miniature microscale engine that can override the role of thermal fluctuations has remained elusive and is an important open challenge. Here we provide the design and theoretical framework for a unique information-based engine - a work-to-work converter - comprising a sub-micron size bead and motor protein-microtubule (MT) complex in an optical trap setup. We demonstrate how by implementing a simple motor protein state-dependent feedback protocol of the optical trap stiffness, this engine is able to harness and convert the movement of a motor protein into work output. Unlike other conventional microengines, the fidelity and performance of this engine is determined by the stochasticity of motor (un)binding characteristics. We obtain an analytical form of the work distribution function, average work output and average power output, providing quantitative predictions for engine performance which are validated by stochastic simulations. Remarkably, the average work output per cycle is at least an order of magnitude higher than the thermal fluctuations and supersedes the performance of other microscale engines realized so far.

physics.bio-ph↗

arXiv2025

A stochastic heat engine driven using a nonlinear protocol

A colloidal particle confined in a time-dependent optical trap can function as a microscopic heat engine, with optimization strategies playing a crucial role in enhancing its performance. In this study, we numerically investigate a Stirling heat engine operating in both passive and active environments using a protocol inspired by the Engineered Swift Equilibration (ESE) method. This approach differs from the standard process and focuses on enhancing engine efficiency, particularly at short time scales. We analyze various fluctuating parameters throughout the cycle to validate the robustness of the engine, and demonstrate a significant enhancement in performance compared to conventional Stirling engines. Most crucially, we observe that the nonlinear protocol can even transform a heat-pump-like operation into a genuine heat engine under strong activity, thereby surpassing bounds imposed on efficiency by high-temperature and quasi-static conditions. Finally, the proposed protocol is designed with experimental feasibility in mind, making it a promising framework for the practical realization of efficient microscopic heat engines.

cond-mat.stat-mech↗

arXiv2025

Differential Characters and $D$-Group Schemes

Let $K$ be a field of characteristic zero with a fixed derivation $\partial$ on it. In the case when $A$ is an abelian scheme, Buium considered the group scheme $K(A)$ which is the kernel of differential characters (also known as Manin characters) on the jet space of $A$. Then $K(A)$ naturally inherits a $D$-group scheme structure. Using the theory of universal vectorial extensions of $A$, he further showed that $K(A)$ is a finite dimensional vectorial extension of $A$. Let $G$ be a smooth connected commutative finite dimensional group scheme over $\mathrm{Spec}~ K$. In this paper, using the theory of differential characters, we show that the associated kernel group scheme $K(G)$ is a finite dimensional $D$-group scheme that is a vectorial extension of such a general $G$. Our proof relies entirely on understanding the structure of jet spaces. Our method also allows us togive a classification of the module of differential characters $\mathbf{X}_\infty(G)$ in terms of primitive characters as a $K\{\partial\}$-module.

math.AG↗

arXiv2025

Stochastic Heat Engine Using a Single Brownian Ellipsoid

Optical tweezers can confine position as well as orientation of a Brownian particle by simultaneously exerting restoring force and torque on it. Here we have proposed the theoretical model of a microscopic Stirling engine, using a passive Brownian ellipsoid as its working substance. The position and the orientation degrees of freedom (DoF) of the ellipsoid in two dimensions (2D), both being confined harmonically by the tweezers, are coupled to a hot and a cold thermal bath time-periodically. The stiffness of the force confinement is also time-periodic such that it resembles a piston-like protocol which drives the Brownian ellipsoid through the strokes of a Stirling cycle. The ellipsoid takes heat from the hot bath and partially converts it into useful thermodynamic work. The extracted work and input heat shows explicit dependence on the shape of the working substance as well as its orientational bias. The operational characteristics of the anisotropic Stirling engine is analyzed using the variance in work and efficiency (in the quasi-static regime), where the latter is bounded by both the Carnot limit as well as the isotropic benchmark. Several ways have been proposed to yield maximum efficiency at a minimum fluctuation in the output. The dissipative coupling between the position and orientation of the ellipsoid, that arises due to its spherical-asymmetry (or, shape anisotropy) and a finite mean orientation, plays an important role to optimize the engine characteristics. Finally, we have analytically explored the slightly anisotropic regime, where the coupling is linearized by suitably tuning the system parameters. The average extracted work has also been calculated in this case, which shows an excellent agreement with the numerical results of the fully anisotropic system, when subjected to the stipulated range of parameters.

cond-mat.stat-mech↗

arXiv2025

Delta Characters and Crystalline Cohomology

The first part of the paper develops the theory of $m$-shifted $π$-typical Witt vectors which can be viewed as subobjects of the usual $π$-typical Witt vectors. We show that the shifted Witt vectors admit a delta structure that satisfy a canonical identity with the delta structure of the usual $π$-typical Witt vectors. Using this theory, we prove that the generalized kernels of arithmetic jet spaces are jet spaces of the kernel at the first level. This also allows us to interpret the arithmetic Picard-Fuchs operator geometrically. For a $π$-formal group scheme $G$, by a previous construction, one attaches a canonical filtered isocrystal $\mathbf{H}_δ(G)$ associated to the arithmetic jet spaces of $G$. In the second half of our paper, we show that $\mathbf{H}_δ(A)$ is of finite rank if $A$ is an abelian scheme. We also prove a strengthened version of a result of Buium on delta characters on abelian schemes. As an application, for an elliptic curve $A$ defined over $\mathbb{Z}_p$, we show that our canonical filtered isocrystal $\mathbf{H}_δ(A) \otimes \mathbb{Q}_p$ is weakly admissible. In particular, if $A$ does not admit a lift of Frobenius, we show that $\mathbf{H}_δ(A) \otimes \mathbb{Q}_p$ is isomorphic to the first crystalline cohomology $\mathbf{H}^1_{\mathrm{cris}}(A) \otimes \mathbb{Q}_p$ in the category of filtered isocrystals. On the other hand, if $A$ admits a lift of Frobenius, then $\mathbf{H}_δ(A) \otimes \mathbb{Q}_p$ is isomorphic to the sub-isocrystal $H^0(A,Ω_A) \otimes \mathbb{Q}_p$ of $\mathbf{H}^1_{\mathrm{cris}}(A) \otimes \mathbb{Q}_p$.

math.NT↗

arXiv2025

Micro Heat Engines With Hydrodynamic Flow

Hydrodynamic flows are often generated in colloidal suspensions. Since colloidal particles are frequently used to construct stochastic heat engines, we study how the hydrodynamic flows influence the output parameters of the engine. We study a single colloidal particle confined in a harmonic trap with time-periodic stiffness that provides the engine protocol, in presence of a steady linear shear flow. The nature of the flow (circular, elliptic or hyperbolic) is externally tunable. At long times, the work done by the flow field is shown to dominate over the thermodynamic (Jarzynski) work done by the trap, if there is an appreciable deviation from the circular flow. The work by the time dependent trap is the sole contributor only for a perfectly circular flow. We also study an extended model, where a microscopic spinning particle (spinor) is tethered close to the colloidal particle, i.e. the working substance of the engine, such that the flow generated by the spinor influences the dynamics of the colloidal particle. We simulate the system and explore the influence of such a flow on the thermodynamics of the engine. We further find that for larger spinning frequencies, the work done by the flow dominates and the system cannot produce thermodynamic work.

cond-mat.stat-mech↗

arXiv2024

Flow Of Information In a Mechanically Quenched Confined Flock

Living entities in a group communicate and transfer information to one another for a variety of reasons. It might be for foraging food, migration, or escaping threats and obstacles, etc. They do so by interacting with each other and also with the environment. The tools from statistical mechanics and information theory can be useful to analyze the flow of information among the living entities modelled as active (i.e. self-propelling) particles. Here we consider the active particles confined in a circular trap. The self-organisation of the particles crucially depends on whether the trap boundary is soft or hard. We quench the trap boundary from soft to hard instantaneously. After the mechanical quench, the particles suddenly find themselves in a hard potential. The self-organised cluster of the active particles, which was stable when the boundary was soft, becomes unstable. The cluster undergoes extreme deformation after the quench to find another stable configuration suitable for the hard potential. Together with the structural relaxation, information regarding the quench also flows throughout the deforming cluster. Here, we quantify the flow of information by computing local transfer entropy. We find that the flow spans the whole cluster, propagating ballistically.

physics.bio-ph↗

arXiv2023

Microscopic Gyration with Dissipative Coupling

Microscopic gyrators, including Brownian gyrators (BGs), require anisotropic fluctuations to perform gyration. It produces a finite current, driving the system out of equilibrium. In a typical BG set-up with an isotropic colloidal particle, the anisotropy sets in by the coupling among space dimensions via an externally applied anisotropic potential confining the particle and the difference between the temperatures along various space dimensions. The coupling is conservative. Here, contrary to a typical BG, first we consider an over-damped, anisotropic colloidal particle (a Brownian ellipsoid), trapped in an isotropic harmonic potential in two dimensions (2D). The space dimensions are coupled by the difference between the longitudinal and transverse frictional drags experienced by the ellipsoid, together with a finite tilt in its orientation due to its chirality. The coupling is dissipative. They are intrinsic properties of the particle. We have shown that this dissipative coupling can generate enough anisotropic fluctuations to perform a steady-state gyration in the Brownian scale. Next, going beyond BG, we have considered an inertial, granular, chiral ellipsoid in 2D, subjected to athermal, anisotropic fluctuations. There is no trapping force confining the granular ellipsoid. However, the coupling between the velocity components of the granular ellipsoid is still dissipative. We have shown that being assisted by the dissipative coupling and the anisotropic fluctuations, the inertial, granular ellipsoid can also perform gyration in 2D. We have also shown that the dominant contribution towards the gyrating frequency can be attributed to the Coriolis force acting on the granular ellipsoid. Hence, the gyrator in the granular scale is also a tiny autonomous machine that generates a directed motion (gyration) from fluctuations. Although there are fundamental differences between the two.

cond-mat.stat-mech↗

arXiv2023

Stochastic Heat Engine Using Multiple Interacting Active Particles

The area of stochastic heat engines using active particles has attracted a lot of attention recently. They have been shown to exhibit advantages over engines using passive particles. In this work, we use multiple self-propelling particles undergoing Vicsek-like aligning interaction as our working system. The particles are confined in a two-dimensional circular trap. The interplay between the confinement and the activity of the particles induces clustering. These clusters change their locations relative to the walls of the trap, when the wall steepness is varied with time. In this work we demonstrate that changing the steepness of the wall and the activity of the particles time-periodically can cause the system to act as an engine. In this setup, we study the variations in extracted work with the activity, rotational diffusion, and the Vicsek radius of individual particles. We also comment on the complications involved in the definition of the engine efficiency in accordance with the usual prescription of stochastic thermodynamics.

cond-mat.stat-mech↗

arXiv2022

Kernel of Arithmetic Jet Spaces

Since the results here have been superseded by another paper cowritten by the author, this article is available for reference purposes only. Fix a Dedekind domain $\mathcal{O}$ and a non-zero prime $\mathfrak{p}$ in it along with a uniformizer $π$. In the first part of the paper, we construct $m$-shifted $π$-typical Witt vectors $W_{mn}(B)$ for any $\mathcal{O}$ algebra $B$ of length $m+n+1$. They are a generalization of the usual $π$-typical Witt vectors. Along with it we construct a lift of Frobenius, called the lateral Frobenius $\tilde{F}: W_{mn}(B) \rightarrow W_{m(n-1)}(B)$ and show that it satisfies a natural identity with the usual Frobenius map. Now given a group scheme $G$ defined over $\mathrm{Spec}~ R$, where $R$ is an $\mathcal{O}$-algebra with a fixed $π$-derivation $δ$ on it, one naturally considers the $n$-th arithmetic jet space $J^nG$ whose points are the Witt ring valued points of $G$. This leads to a natural projection map of group schemes $u: J^{m+n}G \rightarrow J^mG$. Let $N^{mn}G$ denote the kernel of $u$. One of our main results imply that for any $π$-formal group scheme $\hat{G}$ over $\mathrm{Spf}~ R$, $N^{mn}\hat{G}$ is isomorphic to $J^{n-1}(N^{m1}G)$. As an application, if $\hat{G}$ is a smooth commutative $π$-formal group scheme of dimension $d$ and $R$ is of characteristic 0 whose ramification is bounded above by $p-2$, then our result implies that $J^nG$ is a canonical extension of $\hat{G}$ by $(\mathbb{W}_{n-1})^d$ where $\mathbb{W}_{n-1}$ is the $π$-formal group scheme $\hat{\mathbb{A}}^n$ endowed with the group law of addition of Witt vectors. Our results also give a geometric characterization of $G(π^{n+1}R)$ which is the subgroup of points of $G(R)$ that reduces to identity under the modulo $π^{n+1}$ map.

math.AG↗

arXiv2022

Delta Theory of Anderson Modules I: Differential Characters

In this article we develop the theory of differential or delta characters (the arithmetic analogue of Manin characters) of Anderson modules. Here we generalize the construction by Borger and Saha of the canonical finite rank $R$-module $\mathbf{H}(E)$ with a semilinear operator on it to any Anderson module $E$, where $R$ is the base ring which is a $\pi$-adically complete discrete valuation ring with a fixed lift of Frobenius $\phi$ on it. Then we show that $\mathbf{H}(E)$ admits a functorial map to the de Rham cohomology $\mathbf{H}_{\mathrm{dR}}^*(E)$ of $E$ which also preserves the Hodge filtration. We also prove that the module of delta characters $\mathbf{X}_{\infty}(E)$ is finite and free as an $R\{\phi^{*}\}$-module. This leads to a strengthened version of an analogous result by Buium on the generation of differential characters of abelian varieties. We also construct a family of differential modular functions that play the analogous role of $f_{\mathrm{jet}}$ constructed by Buium for elliptic curves. In a subsequent article, the finite rank $R$-module $\mathbf{H}(E)$ will lead to the construction of a canonical $z$-isocrystal $\mathbf{H}_{\delta}(E)$ with a Hodge-Pink filtration on it and we will show that $\mathbf{H}_{\delta}(E)$ is an admissible $z$-isocrystal.

math.NT↗

arXiv2022

Canonical Witt formal scheme extensions and p-torsion groups

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.

math.NT↗

arXiv2022

Exactly solvable model of a passive Brownian heat engine and its comparison with active engines

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.

cond-mat.stat-mech↗