Energetics of a Microscopic Feynman Ratchet Bart Cleuren1and Ralf Eichhorn2 1UHasselt Faculty of Sciences Theory Lab Agoralaan 3590 Diepenbeek Belgium

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Energetics of a Microscopic Feynman Ratchet

Bart Cleuren1and Ralf Eichhorn2

1UHasselt, Faculty of Sciences, Theory Lab, Agoralaan, 3590 Diepenbeek, Belgium

2Nordita, Royal Institute Technology and Stockholm University,

Hannes Alfv´ens v¨ag 12 , SE-106 91 Stockholm, Sweden

E-mail: bart.cleuren@uhasselt.be, eichhorn@nordita.org

Abstract. A general formalism is derived describing both dynamical and energetic

properties of a microscopic Feynman ratchet. Work and heat ﬂows are given as a

series expansion in the thermodynamic forces, obtaining analytical expressions for the

(non)linear response coeﬃcients. Our results extend previously obtained expressions

in the context of a chiral heat pump.

PACS numbers: 05.70.Ln, 05.40.-a, 05.20.-y

Submitted to: J. Stat. Mech.

Contents

1 Introduction 2

2 Microscopic Feynman Ratchet 3

3 Energetics and thermodynamics: general framework 7

4 Energetics and thermodynamics: results and discussion 8

5 Conclusion 11

Acknowledgments 11

arXiv:2210.14647v1 [cond-mat.stat-mech] 26 Oct 2022

CONTENTS 2

1. Introduction

The Feynman ratchet is a celebrated example of a thermal engine [1]. Designed to

illustrate the second law of thermodynamics, it is capable to extract useful work from a

heat ﬂow between two thermal reservoirs at diﬀerent temperatures. The engine consists

of two rigidly connected parts, a ratchet and pawl at one end of a rotational axle, and

vanes at the other end. Each part is immersed in a surrounding gas, and collisions

between the gas particles and the device cause a random rotational motion. The shape

of the ratchet introduces a rotational asymmetry which leads to a systematic rotation

in a certain direction [2], and is driven by the exchange of heat from the hot to the cold

reservoir. Using this rotation for example to lift a weight against the force of gravity, the

whole setup can deliver useful work. On the basis of hand-waving arguments, Feynman

claims in his Lectures on Physics that the engine is capable of operating at Carnot

eﬃciency. It was realised later [3] that the eﬃciency inevitably has to be lower, as the

engine is at all times in simultaneous contact with the two heat reservoirs, resulting in

a heat leakage.

A microscopic analysis of the dynamics and energetics of the original engine has

proven to be notoriously diﬃcult. In part this is caused by the unavoidable recollisions

of the gas particles with the engine giving rise to correlations. Various approaches

have been considered to circumvent these diﬃculties, for example by modelling the

collisions by Langevin noise (eg. [3, 4]) or by considering a discrete setup, eg. [5, 6].

An alternative approach was developed by Van den Broeck and co-workers in a series

of papers in which they meticulously stripped down the engine to its basic constitutes

[7, 8, 9, 10, 11], see also [12]. Considering ideal gases and convex engine parts (which

replace the vanes, ratchet and pawl) an exact microscopic description is derived which

is centred around the stochastic time evolution of the (angular) velocity of the device.

Analytical expressions for the average velocity are obtained in the form of a series

expansion in m/M, the mass ratio between gas particles and device. Even in such an

ideal situation, the interplay between the geometry and the collisions is intricate and

it is not at all obvious in which direction the engine will turn. A description of the

energetics was obtained by adding a torque [13, 14]. In those papers the work and

heat ﬂows were derived from expressions of the average velocity in the linear regime

and making use of the Onsager symmetry. The purpose of the present work is to go

beyond the linear regime by setting the heat and work variables on the same footing as

the angular velocity. Such an extended framework allows to calculate the average (and

higher moments of) work and heat up to any order of the thermodynamic forces.

The paper is organized as follows. Section 2 introduces the microscopic Feynman

ratchet and describes the dynamics of the engine as the result of the external torque

and collisions with the gas particles. Section 3 extends this framework by incorporating

both work and heat variables. The main results are presented in section 4, clarifying

the inﬂuence of the geometry on the motion of the rotor. Finally, we brieﬂy conclude

in section 5.

CONTENTS 3

Figure 1. (a) Sketch of the system under consideration: the engine consists of two

convex objects rigidly connected by an axle, which serves as the ﬁxed rotational axis.

Mounted along the axle is a weight, which is lifted as a result of the collisions of the

gas particles with the convex objects. (b) Each convex object is homogeneous along

the rotational axis, which allows for a two dimensional analysis. Shown is a top view of

the object, indicating the geometrical parameters necessary to describe the collisions

with the gas particles.

2. Microscopic Feynman Ratchet

The original Feynman ratchet was constructed by using a ratchet and pawl on one

end of a rigid rotational axle and a set of vanes on the other end. Because of the

spatial complexity of such a construction, colliding gas particles are very likely to

collide with the various engine parts more than once. This introduces correlations,

and hence a memory, making analytical calculations overwhelmingly complicated. In

order to eliminate such recollisions, and in eﬀect make individual collisions to occur

independently, we reduce the setup to its essence [7]. The purpose of the vanes is to

allow transfer of energy from the gas particles to the engine and vise versa. The ratchet

and pawl set-up introduces a spatial asymmetry which ensures a diﬀerence between

clockwise/anti-clockwise rotation of the engine. A conceptually simple construction

with these properties is shown in ﬁgure 1(a). It consists of two objects rigidly connected

by the rotational axis. For such a construction recollisions can be reduced strongly (so

that they can be neglected in a subsequent analysis) by requiring (i) the shape of each

object to be convex and (ii) the total mass Mof the engine to be much larger than the

mass mof the gas particles. The ﬁrst requirement is purely geometrical and ensures that

a gas particle is directed away from the object after colliding. The second requirement

ensures that a particle, again after collision, is not caught up by the engine. Each object

is surrounded by an ideal gas at a certain temperature and density. In the context of

a heat engine one of the surrounding ideal gases is labeled as the cold reservoir (sub-

or superscript c in the calculations below) and the other one is then the hot reservoir

(sub- or superscript h). A rotational asymmetry is implemented by shaping the objects

asymmetrically with respect to their points of rotation (see Fig.1(b)), and by arranging

them in an asymmetric fashion with respect to each other. Finally the last ingredient

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摘要：

EnergeticsofaMicroscopicFeynmanRatchetBartCleuren1andRalfEichhorn21UHasselt,FacultyofSciences,TheoryLab,Agoralaan,3590Diepenbeek,Belgium2Nordita,RoyalInstituteTechnologyandStockholmUniversity,HannesAlfvensvag12,SE-10691Stockholm,SwedenE-mail:bart.cleuren@uhasselt.be,eichhorn@nordita.orgAbstract.Ag...

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Energetics of a Microscopic Feynman Ratchet Bart Cleuren1and Ralf Eichhorn2 1UHasselt Faculty of Sciences Theory Lab Agoralaan 3590 Diepenbeek Belgium

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