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SECOND LAW OF THERMODYNAMICS: STATUS AND CHALLENGES Date: 14–15 June 2011 Location: San Diego, (USA) ISBN: 978-0-7354-0985-9 Editor(s): Daniel P. Sheehan

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PREFACE: SECOND LAW OF THERMODYNAMICS: STATUS AND CHALLENGES

Daniel P. Sheehan

AIP Conf. Proc. 1411, pp. 1-3; doi:http://dx.doi.org/10.1063/1.3665227 (3 pages)

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Abstract Unavailable
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05.70.Ce Thermodynamic functions and equations of state
03.65.Ca Formalism
81.05.Mh Cermets, ceramic and refractory composites

Reduced statistical fluctuations of the position of an object partitioning in two its environment

Eugenio DelRe, Paolo Di Porto, Stefano Di Sabatino, and Bruno Crosignani

AIP Conf. Proc. 1411, pp. 7-26; doi:http://dx.doi.org/10.1063/1.3665228 (20 pages)

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Through hard‐disk simulations and theoretical considerations on the movement of an object that partitions a microtubule filled with small particles, we find that the vibrations typical of thermal equilibrium are reached after a time that increases exponentially with the number of particles involved. The result is a mechanism capable of breaching, on accessible time scales, the ergodic constraints in nano‐scale systems.
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02.50.Fz Stochastic analysis
45.50.Dd General motion
02.60.Cb Numerical simulation; solution of equations
03.50.De Classical electromagnetism, Maxwell equations

Work from the Most Probable Macrostate, and Relation to the Adiabatic Piston Problem

Jack Denur

AIP Conf. Proc. 1411, pp. 27-37; doi:http://dx.doi.org/10.1063/1.3665229 (11 pages)

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A system's entropy is maximized not when it is localized in its most probable macrostate, but when it is in its most probable distribution of macrostates. This distribution includes all macrostates, including, albeit typically with much smaller probability than the most probable macrostate, those far removed from the most probable one. It is this distribution, and not the most probable macrostate alone, that characterizes true ther‐modynamic equilibrium. Thus, work, albeit typically only in small amounts, is extractable from a system localized in its most probable macrostate. We demonstrate these points via a simple system. We show that a small amount of work can be extracted from a box of gas in thermal equilibrium with a heat reservoir even if the gas is in its most probable macrostate with exactly half of the gas molecules in both the left and right halves of the box. We then qualitatively consider the relation to the adiabatic piston problem.
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05.70.Ce Thermodynamic functions and equations of state
21.10.Jx Spectroscopic factors and asymptotic normalization coefficients
73.20.Fz Weak or Anderson localization
73.20.Jc Delocalization processes

Feynman's Achilles′ Heel?

Lyndsay G. M. Gordon

AIP Conf. Proc. 1411, pp. 38-45; doi:http://dx.doi.org/10.1063/1.3665230 (8 pages)

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Feynman1 argued that a ratchet of the type used in heavy machinery could be used, in a scaled‐down version, to show that the second law of thermodynamics is not just a statistical law, but has absolute status. Hidden is the assumption that this model represented the completely general example of this device−and that it remained so, across the complete range of magnitudes from macro‐ to micro‐domains. It will be argued here, that Feynman′s example is only one construct of a ratchet and cannot represent the general case. A mechanism, with sufficient difference and confined to the micro‐domain, is the subject of the present discussion. Its unique feature is the compressible nature of the pawl as it librates in pawl‐space. The motions of the gear and the pawl are governed by the energy exchanges between them and with the ambient gas. The average linear pressure on the pawl which varies with the magnitude of pawl‐space is associated with the compressibility of the pawl.
Feynman's objective was to show that the prototypal ratchet is incapable of converting heat into work in a manner contrary to the second law. The dubious nature of his examination is not due to an incorrect analysis of his thought experiment, but lies within the assumption that his example represented the general case. In contrast, the ratchet analyzed here does not conform to the principle of detailed balance and thereby the absolute status of the law remains an open question.
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07.10.Pz Instruments for strain, force, and torque
32.70.Cs Oscillator strengths, lifetimes, transition moments
45.20.dh Energy conservation
34.50.Ez Rotational and vibrational energy transfer

The Second Law of Thermodynamics and the thermo‐charged capacitor

Germano D'Abramo

AIP Conf. Proc. 1411, pp. 46-61; doi:http://dx.doi.org/10.1063/1.3665231 (16 pages)

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Some foundational considerations on the peculiar status of the second law as a law of physics are made. Given the atomic nature of matter, whose behaviour is well described by statistical physics, the second law could not hold unconditionally, but only statistically. It is not an absolute law. We argue that this peculiar condition is the main rationale and motivation for pursuing exploratory research to challenge the second law.
Recently [D'Abramo, Phys. Lett. A 374 (2010) 1801, and D'Abramo, Physica A 390∕3 (2011) 482] the concept of vacuum capacitor spontaneously charged harnessing the heat from a single thermal reservoir at room temperature has been introduced, along with a mathematical description of its functioning and a discussion on the main paradoxical feature that seems to violate the second law of thermodynamics.
Here we briefly review these works. We describe the theoretical and practical possibility of exploiting such a thermo‐charged capacitor as voltage∕current generator: if very weak provisos on the physical characteristics of the capacitor are fulfilled, then a non‐zero current should flow across the device, allowing the generation of potentially usable voltage, current and electric power out of a single thermal source at room temperature. Preliminary results show that the power output is tiny but non‐zero.
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05.70.Ce Thermodynamic functions and equations of state
03.50.De Classical electromagnetism, Maxwell equations
07.20.Dt Thermometers
44.05.+e Analytical and numerical techniques

Experimental Challenge to the Second Law of Thermodynamics in High‐Temperature, Gas‐Surface Reactions

D. P. Sheehan, J. T. Garamella, D. J. Mallin, and W. F. Sheehan

AIP Conf. Proc. 1411, pp. 65-81; doi:http://dx.doi.org/10.1063/1.3665232 (17 pages)

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It has been proposed that differences in the adsorption, desorption, dissociation, and recombination rates of chemical species between surfaces can give rise to steady‐state pressure and temperature gradients within a single blackbody cavity under low‐pressure conditions [1‐5]. Such gradients would implicitly violate the second law of thermodynamics. This paper reports on laboratory tests of this proposal. Low‐pressure molecular hydrogen (P ≤ 30Torr) was found to dissociate and desorb preferentially on rhenium compared with tungsten at elevated temperatures (T ≤ 2100K). Blackbody cavities are being constructed from these metals, and their interior surface temperatures monitored. Steady‐state temperature gradients, due to differential gas‐surface reactions, would signal second law breakdown.
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07.20.Dt Thermometers
03.50.De Classical electromagnetism, Maxwell equations
82.60.Hc Chemical equilibria and equilibrium constants
82.30.Lp Decomposition reactions (pyrolysis, dissociation, and fragmentation)

Transformation of Thermal Energy into Electric Energy via Thermionic Emission of Electrons from Dielectric Surfaces in Magnetic Fields

Alexander Perminov and Alexey Nikulov

AIP Conf. Proc. 1411, pp. 82-100; doi:http://dx.doi.org/10.1063/1.3665233 (19 pages)

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It is shown that electrostatic potentials can be created under equilibrium conditions by the Lorentz force motion of thermionic electrons over dielectric surfaces, setting the stage for a conflict with the second law of thermodynamics. These predicted electrostatic potentials have been demonstrated experimentally. We advance this phenomenon as evidence for the feasibility of direct conversion of thermal energy into electricity in the absence of a temperature difference. This challenge to the second law of thermodynamics is compared with related ones.
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41.20.Cv Electrostatics; Poisson and Laplace equations, boundary-value problems
79.40.+z Thermionic emission
84.60.Rb Thermoelectric, electrogasdynamic and other direct energy conversion
82.45.Fk Electrodes

Superconductor Particles As The Working Media Of A Heat Engine

Peter D. Keefe

AIP Conf. Proc. 1411, pp. 101-121; doi:http://dx.doi.org/10.1063/1.3665234 (21 pages)

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A heat engine is presented in which the working media comprises a multiplicity of mutually isolated particles of Type I superconductor which are selectively processed through H‐T phase space so as to convert a heat influx from a high temperature heat reservoir into a useful work output, wherein no heat is rejected to a low temperature heat reservoir.
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05.70.Ce Thermodynamic functions and equations of state
74.25.Ha Magnetic properties including vortex structures and related phenomena
84.60.Rb Thermoelectric, electrogasdynamic and other direct energy conversion
75.60.Ej Magnetization curves, hysteresis, Barkhausen and related effects

Observations of Persistent Current at Non‐Zero Resistance: Challenge to the Second Law of Thermodynamics

Alexey Nikulov

AIP Conf. Proc. 1411, pp. 122-144; doi:http://dx.doi.org/10.1063/1.3665235 (23 pages)

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This article details why the mesoscopic quantum phenomenon known as persistent current challenges the second law of thermodynamics. The persistent current is an equilibrium phenomenon as real as Nyquist (Johnson) noise, but in contrast, it is not random; its direct component (i.e. zero‐frequency component) is non‐zero because of the discreteness of the permitted state spectrum of electrons in normal metal rings and Cooper pairs in superconductor rings. The persistent current observed in mesoscopic rings with non‐zero resistance is effectively directed Brownian motion, which cannot decay despite its non‐zero energy dissipation. This is due to the equilibration between the dissipative force with the change of angular momentum of electrons (or Cooper pairs), owing to the quantization condition on the wave function describing their states in the ring. The observations of electric potential difference on ring‐halves having persistent current raise the possibility of utilizing persistent currents for useful work, in conflict with the second law.
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05.20.Dd Kinetic theory
05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)
03.50.De Classical electromagnetism, Maxwell equations
02.50.Cw Probability theory

Experimental Measurements of Electric Fields in Diodic Air Gaps: Toward a Second Law Challenge

D. P. Sheehan and J. H. Wright

AIP Conf. Proc. 1411, pp. 147-157; doi:http://dx.doi.org/10.1063/1.3665236 (11 pages)

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Over the last ten years, researchers at USD have investigated a series of challenges to the second law of thermodynamics that involve the exploitation of intense vacuum electric fields generated by solid‐state diodic contacts [1, 2, 3]. Although theoretical arguments and numerical simulations supported the existence of these fields, experimental verification had been lacking. This article reviews the theoretical basis for these diodic electric fields and details recent laboratory experiments that have verified their location, intensity, and rechargability.
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46.55.+d Tribology and mechanical contacts
03.50.De Classical electromagnetism, Maxwell equations
02.60.Cb Numerical simulation; solution of equations
84.37.+q Measurements in electric variables (including voltage, current, resistance, capacitance, inductance, impedance, and admittance, etc.)

Experimental Evidence Violating Laws of Thermodynamics In Magnetostrictive Materials

Gerald Pellegrini

AIP Conf. Proc. 1411, pp. 158-166; doi:http://dx.doi.org/10.1063/1.3665237 (9 pages)

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There have been a number of experimental results that show that the magnetomechanical coupling in certain magnetostrictive materials do not satisfy the thermodynamic Maxwell relations. These startling results have come from different laboratories including the Naval Surface Warfare Center, Carderock Division and the University of Maryland, College Park. If these experiments are correct, and the Maxwell relations are violated, the experiments represent the very important discovery of magnetic materials that do math obey the standard thermodynamic laws.
In this paper, the experimental evidence will be presented along with the theoretical analysis demonstrating that the experimental data is not consistent with the existence of the standard thermodynamic potentials, and demonstrating how magnetomechanical cycles may be constructed by which energy can be extracted from the ambient temperature environment. Experimental measurements of an extraction of energy in such magnetomechanical cycles will also be presented.
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75.80.+q Magnetomechanical effects, magnetostriction
03.50.De Classical electromagnetism, Maxwell equations
75.60.Ej Magnetization curves, hysteresis, Barkhausen and related effects
41.20.Gz Magnetostatics; magnetic shielding, magnetic induction, boundary-value problems

The Proell Effect: A Macroscopic Maxwell's Demon

Kenneth M. Rauen

AIP Conf. Proc. 1411, pp. 167-192; doi:http://dx.doi.org/10.1063/1.3665238 (26 pages)

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Maxwell's Demon is a legitimate challenge to the Second Law of Thermodynamics when the “demon” is executed via the Proell effect. Thermal energy transfer according to the Kinetic Theory of Heat and Statistical Mechanics that takes place over distances greater than the mean free path of a gas circumvents the microscopic randomness that leads to macroscopic irreversibility. No information is required to sort the particles as no sorting occurs; the entire volume of gas undergoes the same transition. The Proell effect achieves quasi‐spontaneous thermal separation without sorting by the perturbation of a heterogeneous constant volume system with displacement and regeneration. The classical analysis of the constant volume process, such as found in the Stirling Cycle, is incomplete and therefore incorrect. There are extra energy flows that classical thermo does not recognize. When a working fluid is displaced across a regenerator with a temperature gradient in a constant volume system, complimentary compression and expansion work takes place that transfers energy between the regenerator and the bulk gas volumes of the hot and cold sides of the constant volume system. Heat capacity at constant pressure applies instead of heat capacity at constant volume. The resultant increase in calculated, recyclable energy allows the Carnot Limit to be exceeded in certain cycles. Super‐Carnot heat engines and heat pumps have been designed and a US patent has been awarded.
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03.50.De Classical electromagnetism, Maxwell equations
05.70.Ln Nonequilibrium and irreversible thermodynamics
07.20.Pe Heat engines; heat pumps; heat pipes
07.20.Dt Thermometers

The Production of Electricity out of a Heat Bath

Roderich W. Graeff

AIP Conf. Proc. 1411, pp. 193-217; doi:http://dx.doi.org/10.1063/1.3665239 (25 pages)

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In order to clarify the dispute between Loschmidt and Boltzmann∕Maxwell concerning the existence of a temperature gradient in insulated vertical columns of gas, liquid or solids, macroscopic measurements of the temperature distribution in air, water and solids were performed. A negative temperature gradient, cold at the top and warm at the bottom, is found in insulated vertical tubes, while the outside environment has a reverse gradient. This is explainable by the influence of gravity. It allows the production of electricity out of a heat bath.
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03.50.De Classical electromagnetism, Maxwell equations
44.25.+f Natural convection
07.20.Dt Thermometers
04.40.Nr Einstein-Maxwell spacetimes, spacetimes with fluids, radiation or classical fields

Charge Acceleration and Field‐Lines Curvature: A Fundamental Symmetry and Consequent Asymmetries

Avshalom C. Elitzur, Eliahu Cohen, and Paz Beniamini

AIP Conf. Proc. 1411, pp. 221-244; doi:http://dx.doi.org/10.1063/1.3665240 (24 pages)

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When a charge accelerates, its field‐lines curve in a typical pattern. This pattern resembles the curvature induced on the field‐lines by a neighboring charge. Not only does the latter case involve a similar curvature, it moreover results in attraction∕repulsion. This suggests a hitherto unnoticed causal symmetry: charge acceleration ⇔ field curvature. We prove quantitatively that these two phenomena are essentially one and the same. The field stores some of the charge's mass, yet it is extended in space, hence when the charge accelerates, inertia makes the field lag behind. The resulting stress in the field stores some of the charge's kinetic energy in the form of potential energy. The electrostatic interaction is the approximate mirror image of this process: The potential energy stored within the field turns into the charge's kinetic energy. This partial symmetry offers novel insights into two debated issues in electromagnetism The question whether a charge radiates in a gravitational field receives a new twist: If all the charge's field‐lines end with opposite charges that also resist gravity, no radiation is expected. Similarly for the famous absence of a physical manifestation of the Maxwell equations′ advanced solution: Just as Einstein argued, the reason for this absence is probabilistic rather than reflecting some inherent time‐asymmetry. Despite the apparent equivalence between the “ontological” and “instrumentalist” viewpoints concerning the physical reality of field‐lines, there may be cases in which their experimental predictions differ.
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05.70.Ln Nonequilibrium and irreversible thermodynamics
41.20.Cv Electrostatics; Poisson and Laplace equations, boundary-value problems
03.50.De Classical electromagnetism, Maxwell equations
45.20.df Momentum conservation

Non‐thermodynamic behavior for non‐ergodic interactions

B. Gaveau and L. S. Schulman

AIP Conf. Proc. 1411, pp. 245-255; doi:http://dx.doi.org/10.1063/1.3665241 (11 pages)

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The phenomenon suggested by the title should surprise no one. What may be surprising is how easy it is to produce a quantum system with this feature; moreover, that system is one that is often used for showing how systems equilibrate. We provide an example where two systems can be brought into repeated contact, not be at the same temperature, but nevertheless no energy passes between them, nor is any work needed maintain disequilibrium.
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05.70.Ce Thermodynamic functions and equations of state
05.45.Xt Synchronization; coupled oscillators
05.30.Jp Boson systems
45.20.dh Energy conservation

Quantum Thermodynamics of Photo and Solar Cells

Konstantin E. Dorfman, Kimberly R. Chapin, C. H. Raymond Ooi, Anatoly A. Svidzinsky, and Marlan O. Scully

AIP Conf. Proc. 1411, pp. 256-264; doi:http://dx.doi.org/10.1063/1.3665242 (9 pages)

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Quantum coherence can increase the quantum efficiency of various thermodynamic systems. For example, we can enhance the quantum efficiency for a quantum dot photocell, a laser based solar cell and the photo‐Carnot quantum heat engine. Our results are fully consistent with the laws of thermodynamics contrary to comments found in the paper of A. P. Kirk, Phys. Rev. Lett. 106, 048703 (2011).
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05.70.Ce Thermodynamic functions and equations of state
85.35.Be Quantum well devices (quantum dots, quantum wires, etc.)
07.20.Pe Heat engines; heat pumps; heat pipes
42.50.Ar Photon statistics and coherence theory

Repelling Point Bosons

J. B. McGuire

AIP Conf. Proc. 1411, pp. 265-274; doi:http://dx.doi.org/10.1063/1.3665243 (10 pages)

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There is a body of conventional wisdom that holds that a solvable quantum problem, by virtue of its solvability, is pathological and thus irrelevant. It has been difficult to refute this view owing to the paucity of theoretical constructs and experimental results. Recent experiments involving equivalent ions trapped in a spatial conformation of extreme anisotropic confinement (longitudinal extension tens, hundreds or even thousands of times transverse extension) have modified the view of relevancy, and it is now possible to consider systems previously thought pathological, in particular point Bosons that repel in one dimension. It has been difficult for the experimentalists to utilize existing theory, mainly due to long‐standing theoretical misunderstanding of the relevance of the permutation group, in particular the non‐commutativity of translations (periodicity) and transpositions (permutation). This misunderstanding is most easily rectified in the case of repelling Bosons.
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05.30.Jp Boson systems
02.50.Cw Probability theory
06.30.Dr Mass and density
02.60.Cb Numerical simulation; solution of equations

On Entropy In Eulerian Thermodynamics

Christian Fronsdal and Abhishek Pathak

AIP Conf. Proc. 1411, pp. 277-291; doi:http://dx.doi.org/10.1063/1.3665244 (15 pages)

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To the student of thermodynamics the most difficult subject is entropy. In this paper we examine the actual, practical application of entropy to two simple systems, the homogeneous slab with fixed boundary values of the temperature, and an isolated atmosphere in the presence of the static gravitational field. The first gives valuable insight into the nature of entropy that is subsequently applied to the second system. It is a basic tenet of thermodynamics that the equilibrium of an extended, homogeneous and isolated system is characterized by a uniform temperature distribution and it is a strongly held belief that this remains true in the presence of gravity. We find that this is consistent with the equations of extended thermodynamics but that entropy enters in an essential way. The principle of equivalence takes on a new aspect.
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05.70.Ce Thermodynamic functions and equations of state
07.20.Dt Thermometers
47.10.ad Navier-Stokes equations
47.85.Dh Hydrodynamics, hydraulics, hydrostatics

I. Time Reversibility Concepts, the Second Law and Irreversible Thermodynamics

Christopher G. Jesudason

AIP Conf. Proc. 1411, pp. 292-307; doi:http://dx.doi.org/10.1063/1.3665245 (16 pages)

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Time reversibility concepts and transformations are first reviewed and difficulties with the standard formulations indicated. The kinetic equations which were constructed to exhibit reciprocity relations in their transition probabilities based on time reversal ideas are examined next and a first principle analysis shows that the standard forms are not in accord with the first principles. A thermodynamical theory based on the Kelvin‐Clausius‐Planck definition of entropy and a modified form of the Benofy and Quay postulate concerning conductive heat is developed and reciprocity and other relations are derived as an example of one possible alternative to the standard treatments with their indicated inconsistencies.
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03.50.De Classical electromagnetism, Maxwell equations
45.20.df Momentum conservation
45.50.Dd General motion
02.50.Cw Probability theory

II. The Second Law in Relation to Thermal Radiative Transfer

Christopher G. Jesudason

AIP Conf. Proc. 1411, pp. 308-326; doi:http://dx.doi.org/10.1063/1.3665246 (19 pages)

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Planck introduced the quantum hypothesis from his Blackbody radiation studies, where he and subsequent workers opined that classical mechanics and electrodynamical theories could not account for the phenomenon. Hence a statistical mechanics with an appropriate Second law entropy was invented and coupled to the First law to account for quantum effects. Here, as an academic exercise we derive the quantum of energy by considering two structures, that of the dipole oscillators on a 2‐D surface and the scattering of radiation into the 3‐D cavity. Previous derivations are briefly cited and reviewed where none followed this approach. One prediction from this first order Brownian motion development is that a 2‐D sheet of oscillators should emit radiation largely with energy density factor T1 of the Kelvin temperature T, rather than that deduced as T4 from detailed balance. Preliminary measurements conducted here seemed to verify the the T1 density. The first order theory also admits a possibility of nonlinear quanta and the consequences are explored briefly. It was noticed in passing during the experimentation that certain bodies suspended in a vacuum exhibited small persistent temperature differentials. A Second law statement is presented for such cases and consequences explored for processes that are not coupled by Newtonian momentum energy transfer mechanisms, such as for the radiation field as deduced by Planck. The different forms of heat transfer due to different laws (e.g. gravity waves and electromagnetic waves) are strictly separable and cannot be confused or forced to an equivalence. We generalize on the Zeroth law, the Kirchoff law and postulate an appropriate entropy form due to these generalizations.
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44.40.+a Thermal radiation
03.50.De Classical electromagnetism, Maxwell equations
05.45.Xt Synchronization; coupled oscillators
05.40.Jc Brownian motion

Revisiting The Second Law of Energy Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization

Milivoje M. Kostic

AIP Conf. Proc. 1411, pp. 327-350; doi:http://dx.doi.org/10.1063/1.3665247 (24 pages)

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Sadi Carnot's ingenious reasoning of reversible cycles (1824) laid foundations for The Second Law before The First Law of energy conservation was even known (Joule 1843) and long before Thermodynamic concepts were established in 1850s. A century later, Bridgman (1941) ‘complained’ that “there are almost as many formulations of The Second Law as there have been discussions of it.” Even today, The Second Law remains so obscure, due to the lack of its comprehension, that it continues to attract new efforts at clarification, including this one.
The Laws of Thermodynamics have much wider, including philosophical significance and implication, than their simple expressions based on the experimental observations—they are The Fundamental Laws of Nature: The Zeroth (equilibrium existentialism), The First (conservational transformationalism), The Second (irreversible directional transformationalism), and The Third (unattainability of emptiness). They are defining and unifying our comprehension of all existence and transformations in the universe. The forces, due to non‐equilibrium of mass‐energy in space (non‐uniform ‘concentrations’), causing the mass‐energy displacement, thus defining the process direction, are manifested by tendency of mass‐energy transfer in time towards common equilibrium—cause‐and‐effect forced tendency of equi‐partition of mass‐energy. It should not be confused with local creation of non‐equilibrium and∕or ‘organized structures’ on expense of ‘over‐all’ non‐equilibrium, by spontaneous and irreversible conversion (dissipation) of other energy forms into the thermal energy, always and everywhere accompanied with entropy generation (randomized equi‐partition of energy per absolute temperature level).
The fundamental laws of nature are considered to be axiomatic and many believe they could not be explained, proven or questioned. However, everything may and should be questioned, reasoned, explained and possibly proven. The miracles are until they are comprehended and understood.
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05.70.Ln Nonequilibrium and irreversible thermodynamics
07.20.Pe Heat engines; heat pumps; heat pipes
44.20.+b Boundary layer heat flow
45.20.dh Energy conservation

Large (and Small) Energy Fluctuations in a Single Classical Degree of Freedom and the Second Law of Thermodynamics

Jack Denur

AIP Conf. Proc. 1411, pp. 351-356; doi:http://dx.doi.org/10.1063/1.3665248 (6 pages)

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Energy fluctuations in a single classical degree of freedom above the ground state at thermodynamic equilibrium at temperature T are typically of average magnitude ∼ kBT. However, we show that the average magnitude of such fluctuations can be much larger (or much smaller) than kBT, indeed, that at least in principle it can be infinite (or arbitrarily close to 0). Nevertheless, the average energy fluctuation magnitude being untypically large (or untypically small) does not violate the second law of thermodynamics. For, if the average magnitude of energy fluctuations is much larger than kBT, then particle motion along the degree of freedom must manifest extreme spatial delocalization. The cost of locating the fluctuating particle along its degree of freedom equals or exceeds the large energy gain obtained upon finding it with an energy of much more than kBT above its ground state. The particle loses as much or more ability to do work via its spatial delocalization than it gains via the energy fluctuation. Similarly, if the average magnitude of energy fluctuations is much smaller than kBT, then the small energy yield obtainable upon locating the particle is compensated for by the small cost of locating it.
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05.70.Ce Thermodynamic functions and equations of state
73.20.Jc Delocalization processes
03.65.Ge Solutions of wave equations: bound states
31.50.Bc Potential energy surfaces for ground electronic states
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