Entropy

254 KiB

Open AccessArticle

by Rémi Léandre

Entropy 2004, 6(1), 180-215; https://doi.org/10.3390/e6010180 - 31 Mar 2004

Cited by 5 | Viewed by 5726

We define heat kernel measure on punctured spheres. The random field which is got by this procedure is not Gaussian. We define a stochastic line bundle on the loop space, such that the punctured sphere corresponds to a generalized parallel transport on this [...] Read more.

We define heat kernel measure on punctured spheres. The random field which is got by this procedure is not Gaussian. We define a stochastic line bundle on the loop space, such that the punctured sphere corresponds to a generalized parallel transport on this line bundle. Markov property along the sewing loops corresponds to an operadic structure of the stochastic W.Z.N.W. model. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

39 KiB

Open AccessEditorial

Special Issue on Quantum Limits to the Second Law of Thermodynamics

by Alexey V. Nikulov and Daniel P. Sheehan

Entropy 2004, 6(1), 1-10; https://doi.org/10.3390/e6010001 - 31 Mar 2004

Cited by 17 | Viewed by 6364

Abstract

Over fifty years ago Arthur Eddington wrote [1]: “The second law of thermodynamics holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's [...] Read more.

Over fifty years ago Arthur Eddington wrote [1]: “The second law of thermodynamics holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation, well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation”.[...] Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

145 KiB

Open AccessArticle

A Collision Between Dynamics and Thermodynamics

by Craig Callender

Entropy 2004, 6(1), 11-20; https://doi.org/10.3390/e6010011 - 30 Mar 2004

Cited by 10 | Viewed by 6216

Abstract

Philosophers of science have found the literature surrounding Maxwell's demon deeply problematic. This paper explains why, summarizing various philosophical complaints and adding to them. The first part of the paper critically evaluates attempts to exorcise Maxwell's demon; the second part raises foundational questions [...] Read more.

Philosophers of science have found the literature surrounding Maxwell's demon deeply problematic. This paper explains why, summarizing various philosophical complaints and adding to them. The first part of the paper critically evaluates attempts to exorcise Maxwell's demon; the second part raises foundational questions about some of the putative demons that are being summoned. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

99 KiB

Open AccessArticle

A Modified Szilard's Engine: Measurement, Information, and Maxwell's Demon

by Michael Devereux

Entropy 2004, 6(1), 102-115; https://doi.org/10.3390/e6010102 - 23 Mar 2004

Cited by 1 | Viewed by 7426

Abstract

Using an isolated measurement process, I've calculated the effect measurement has on entropy for the multi-cylinder Szilard engine. This calculation shows that the system of cylinders possesses an entropy associated with cylinder total energy states, and that it records information transferred at measurement. [...] Read more.

Using an isolated measurement process, I've calculated the effect measurement has on entropy for the multi-cylinder Szilard engine. This calculation shows that the system of cylinders possesses an entropy associated with cylinder total energy states, and that it records information transferred at measurement. Contrary to other's results, I've found that the apparatus loses entropy due to measurement. The Second Law of Thermodynamics may be preserved if Maxwell's demon gains entropy moving the engine partition. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

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946 KiB

Open AccessArticle

Delocalization and Sensitivity of Quantum Wavepacket in Coherently Perturbed Kicked Anderson Model

by Hiroaki Yamada

Entropy 2004, 6(1), 133-152; https://doi.org/10.3390/e6010133 - 21 Mar 2004

Viewed by 4200

Abstract

We consider quantum diffusion of the initially localized wavepacket in one-dimensional kicked disordered system with classical coherent perturbation. The wavepacket localizes in the unperturbed kicked Anderson model. However, the wavepacket get delocalized even by coupling with monochromatic perturbation. We call the state "dynamically [...] Read more.

We consider quantum diffusion of the initially localized wavepacket in one-dimensional kicked disordered system with classical coherent perturbation. The wavepacket localizes in the unperturbed kicked Anderson model. However, the wavepacket get delocalized even by coupling with monochromatic perturbation. We call the state "dynamically delocalized state". It is numerically shown that the delocalized wavepacket spread obeying diffusion law, and the perturbation strength dependence of the diffusion rate is given. The sensitivity of the delocalized state is also shown by the time-reversal experiment after random change in phase of the wavepacket. Moreover, it is found that the diffusion strongly depend on the initial phase of the perturbation. We discuss a relation between the "classicalization" of the quantum wave packet and the time-dependence of the initial phase dependence. The complex structure of the initial phase dependence is related to the entropy production in the quantum system. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

28 KiB

Open AccessArticle

Smoluchowski's Trapdoor

by Lyndsay G.M. Gordon

Entropy 2004, 6(1), 96-101; https://doi.org/10.3390/e6010096 - 21 Mar 2004

Cited by 4 | Viewed by 6909

Abstract

A mechanism of a gated pore in a membrane is described. Fluxes of gas molecules pass through the pore and produce a pressure gradient in a process that challenges the second law. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

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38 KiB

Open AccessArticle

A Maxwellian Valve based on centrifugal forces

by Lyndsay G.M. Gordon

Entropy 2004, 6(1), 87-95; https://doi.org/10.3390/e6010087 - 21 Mar 2004

Cited by 6 | Viewed by 5094

Abstract

A mechanism is described which can create a chemical potential gradient from a single heat reservoir. The mechanism is equivalent to a Maxwellian valve. Heat supplies the energy through thermal fluctuations to form the gradient contrary to the second law. A quantitative analysis [...] Read more.

A mechanism is described which can create a chemical potential gradient from a single heat reservoir. The mechanism is equivalent to a Maxwellian valve. Heat supplies the energy through thermal fluctuations to form the gradient contrary to the second law. A quantitative analysis of the system is given. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

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74 KiB

Open AccessArticle

The Decrease in Entropy via Fluctuations

by Lyndsay G.M. Gordon

Entropy 2004, 6(1), 38-49; https://doi.org/10.3390/e6010038 - 21 Mar 2004

Cited by 10 | Viewed by 5836

Abstract

Classical and quantum aspects of fluctuations are reviewed in a discussion on the universality of the second law. Consideration is given to the need of information and the requirement of a ratchet and pawl-type mechanism for the utilization of energy from fluctuations. Some [...] Read more.

Classical and quantum aspects of fluctuations are reviewed in a discussion on the universality of the second law. Consideration is given to the need of information and the requirement of a ratchet and pawl-type mechanism for the utilization of energy from fluctuations. Some aspects of the quantum theory of the Copenhagen school are compared to those of Bohm within the discussion. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

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Figure 1
The device of Szilard. (A) The load has been coupled to the piston by an intelligent being and simultaneously the trapdoor has been closed. (B) Work has been accomplished by the gas expansion. (C) The trapdoor has been opened and simultaneously the load has been uncoupled and supported. (D) The piston has been moved to a central position. The cycle recommences at A with the gas molecule on the same or opposite side of the piston. Full article ">Figure 2
The device of Popper as perceived by the author<a href="#fn001-entropy-06-00038" class="html-fn">1</a>. The same operations as in Szilard’s device (<a href="#entropy-06-00038-f001" class="html-fig">Fig. 1</a>) except a converter is added to replace the intelligent being. The Plate P attached to the piston rod pushes or pulls the plates fixed to the converter belt. (See the text for details.) Full article ">Figure 3
A device as described in <a href="#entropy-06-00038-f002" class="html-fig">Fig. 2</a> but with an additional wheel that is fixed to the axle of a converter wheel and makes a second linkage (one of control via a pawl and trapdoor) between the converter and the Szilard engine. The first linkage is via the piston rod. Full article ">Figure 4
Plan view of the device illustrated in <a href="#entropy-06-00038-f003" class="html-fig">Fig. 3</a>. The pawl and the trapdoor are fixed to the same axle lying vertically in the plane of the paper. The pawl is engaged, the trapdoor is open (not shown) at indicated position. Full article ">

138 KiB

Open AccessArticle

Thomson's formulation of the second law for macroscopic and finite work sources

by Armen E. Allahverdyan, Roger Balian and Theo M. Nieuwenhuizen

Entropy 2004, 6(1), 30-37; https://doi.org/10.3390/e6010030 - 18 Mar 2004

Cited by 5 | Viewed by 5821

Abstract

Thomson's formulation of the second law states: no work can be extracted from an equilibrium system through a cyclic process. A simple, general proof is presented for the case of macroscopic sources of work. Next the setup is generalized towards situations, where the [...] Read more.

Thomson's formulation of the second law states: no work can be extracted from an equilibrium system through a cyclic process. A simple, general proof is presented for the case of macroscopic sources of work. Next the setup is generalized towards situations, where the corresponding work-source is not macroscopic. It is shown that using such a source one can extract energy from an equilibrium system by means of a cyclic process. However, this extraction is accompanied by an entropy increase of the source, in a manner resembling the Clausius inequality. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

256 KiB

Open AccessArticle

On Expansion of a Spherical Enclosure Bathed in Zero-Point Radiation

by Jirí J. Mares, Václav Spicka, Jozef Kristofik and Pavel Hubik

Entropy 2004, 6(1), 216-222; https://doi.org/10.3390/e6010216 - 17 Mar 2004

Viewed by 5464

Abstract

In the present contribution a simple thought experiment made with an idealized spherical enclosure bathed in zero-point (ZP) electromagnetic radiation and having walls made of a material with an upper frequency cut-off has been qualitatively analysed. As a result, a possible mechanism of [...] Read more.

In the present contribution a simple thought experiment made with an idealized spherical enclosure bathed in zero-point (ZP) electromagnetic radiation and having walls made of a material with an upper frequency cut-off has been qualitatively analysed. As a result, a possible mechanism of filling real cavities with ZP radiation based on Doppler's effect has been suggested and corresponding entropy changes have been discussed. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

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171 KiB

Open AccessArticle

Second Law Violation By Magneto-Caloric Effect Adiabatic Phase Transition of Type I Superconductive Particles

by Peter Keefe

Entropy 2004, 6(1), 116-127; https://doi.org/10.3390/e6010116 - 17 Mar 2004

Cited by 13 | Viewed by 6733

Abstract

The nature of the thermodynamic behavior of Type I superconductor particles, having a cross section less than the Ginzburg-Landau temperature dependent coherence length is discussed for magnetic field induced adiabatic phase transitions from the superconductive state to the normal state. Argument is advanced [...] Read more.

The nature of the thermodynamic behavior of Type I superconductor particles, having a cross section less than the Ginzburg-Landau temperature dependent coherence length is discussed for magnetic field induced adiabatic phase transitions from the superconductive state to the normal state. Argument is advanced supporting the view that when the adiabatic magneto-caloric process is applied to particles, the phase transition is characterized by a decrease in entropy in violation of traditional formulations of the Second Law, evidenced by attainment of a final process temperature below that which would result from an adiabatic magneto-caloric process applied to bulk dimensioned specimens. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

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150 KiB

Open AccessArticle

Internal Structure of Elementary Particle and Possible Deterministic Mechanism of Biological Evolution

by Alexei V. Melkikh

Entropy 2004, 6(1), 223-232; https://doi.org/10.3390/e6010223 - 16 Mar 2004

Cited by 2 | Viewed by 7147

Abstract

The possibility of a complicated internal structure of an elementary particle was analyzed. In this case a particle may represent a quantum computer with many degrees of freedom. It was shown that the probability of new species formation by means of random mutations [...] Read more.

The possibility of a complicated internal structure of an elementary particle was analyzed. In this case a particle may represent a quantum computer with many degrees of freedom. It was shown that the probability of new species formation by means of random mutations is negligibly small. Deterministic model of evolution is considered. According to this model DNA nucleotides can change their state under the control of elementary particle internal degrees of freedom. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

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414 KiB

Open AccessArticle

A New Thermodynamics from Nuclei to Stars

by Dieter H.E. Gross

Entropy 2004, 6(1), 158-179; https://doi.org/10.3390/e6010158 - 16 Mar 2004

Cited by 21 | Viewed by 7900

Abstract

Equilibrium statistics of Hamiltonian systems is correctly described by the microcanonical ensemble. Classically this is the manifold of all points in the N-body phase space with the given total energy. Due to Boltzmann's principle, e^S=tr(δ(E-H)), its geometrical size is related to [...] Read more.

Equilibrium statistics of Hamiltonian systems is correctly described by the microcanonical ensemble. Classically this is the manifold of all points in the N-body phase space with the given total energy. Due to Boltzmann's principle, e^S=tr(δ(E-H)), its geometrical size is related to the entropy S(E,N,...). This definition does not invoke any information theory, no thermodynamic limit, no extensivity, and no homogeneity assumption, as are needed in conventional (canonical) thermo-statistics. Therefore, it describes the equilibrium statistics of extensive as well of non-extensive systems. Due to this fact it is the fundamental definition of any classical equilibrium statistics. It can address nuclei and astrophysical objects as well. All kind of phase transitions can be distinguished sharply and uniquely for even small systems. It is further shown that the second law is a natural consequence of the statistical nature of thermodynamics which describes all systems with the same -- redundant -- set of few control parameters simultaneously. It has nothing to do with the thermodynamic limit. It even works in systems which are by far than any thermodynamic "limit". Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

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108 KiB

Open AccessArticle

Entropic localization in non-unitary Newtonian gravity

by Sergio Filippo De and Filippo Maimone

Entropy 2004, 6(1), 153-157; https://doi.org/10.3390/e6010153 - 16 Mar 2004

Cited by 6 | Viewed by 5180

Abstract

The localizing properties and the entropy production of the Newtonian limit of a nonunitary version of fourth order gravity are analyzed. It is argued that pure highly unlocalized states of the center of mass motion of macroscopic bodies rapidly evolve into unlocalized ensembles [...] Read more.

The localizing properties and the entropy production of the Newtonian limit of a nonunitary version of fourth order gravity are analyzed. It is argued that pure highly unlocalized states of the center of mass motion of macroscopic bodies rapidly evolve into unlocalized ensembles of highly localized states. The localization time and the final entropy are estimated. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

73 KiB

Open AccessArticle

A Concise Equation of State for Aqueous Solutions of Electrolytes Incorporating Thermodynamic Laws and Entropy

by Raji Heyrovská

Entropy 2004, 6(1), 128-132; https://doi.org/10.3390/e6010128 - 16 Mar 2004

Cited by 4 | Viewed by 6054

Abstract

Recently, the author suggested a simple and composite equation of state by incorporating fundamental thermodynamic properties like heat capacities into her earlier concise equation of state for gases based on free volume and molecular association / dissociation. This work brings new results for [...] Read more.

Recently, the author suggested a simple and composite equation of state by incorporating fundamental thermodynamic properties like heat capacities into her earlier concise equation of state for gases based on free volume and molecular association / dissociation. This work brings new results for aqueous solutions, based on the analogy of the equation of state for gases and solutions over wide ranges of pressures (for gases) and concentrations (for solutions). The definitions of entropy and heat energy through the equation of state for gases, also holds for solutions. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

240 KiB

Open AccessArticle

Modified Feynman ratchet with velocity-dependent fluctuations

by Jack Denur

Entropy 2004, 6(1), 76-86; https://doi.org/10.3390/e6010076 - 15 Mar 2004

Cited by 7 | Viewed by 7924

Abstract

The randomness of Brownian motion at thermodynamic equilibrium can be spontaneously broken by velocity-dependence of fluctuations, i.e., by dependence of values or probability distributions of fluctuating properties on Brownian-motional velocity. Such randomness-breaking can spontaneously obtain via interaction between Brownian-motional Doppler effects --- which [...] Read more.

The randomness of Brownian motion at thermodynamic equilibrium can be spontaneously broken by velocity-dependence of fluctuations, i.e., by dependence of values or probability distributions of fluctuating properties on Brownian-motional velocity. Such randomness-breaking can spontaneously obtain via interaction between Brownian-motional Doppler effects --- which manifest the required velocity-dependence --- and system geometrical asymmetry. A non random walk is thereby spontaneously superposed on Brownian motion, resulting in a systematic net drift velocity despite thermodynamic equilibrium. The time evolution of this systematic net drift velocity --- and of velocity probability density, force, and power output --- is derived for a velocity-dependent modification of Feynman's ratchet. We show that said spontaneous randomness-breaking, and consequent systematic net drift velocity, imply: bias from the Maxwellian of the system's velocity probability density, the force that tends to accelerate it, and its power output. Maximization, especially of power output, is discussed. Uncompensated decreases in total entropy, challenging the second law of thermodynamics, are thereby implied. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

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179 KiB

Open AccessArticle

From Randomness to Order

by Jorge Berger

Entropy 2004, 6(1), 68-75; https://doi.org/10.3390/e6010068 - 12 Mar 2004

Cited by 2 | Viewed by 6023

Abstract

I review some selected situations in which order builds up from randomness, or a losing trend turns into winning. Except for Section 4 (which is mine), all cases are well documented and the price paid to achieve order is apparent. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

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585 KiB

Open AccessArticle

Langevin approach to the Porto system

by Jirí Bok and Vladislav Cápek

Entropy 2004, 6(1), 57-67; https://doi.org/10.3390/e6010057 - 12 Mar 2004

Cited by 8 | Viewed by 5361

Abstract

M. Porto (Phys. Rev. E 63 (2001) 030102) suggested a system consisting of Coulomb interacting particles, forming a linear track and a rotor, and working as a molecular motor. Newton equations with damping for the rotor coordinate on the track x, with a [...] Read more.

M. Porto (Phys. Rev. E 63 (2001) 030102) suggested a system consisting of Coulomb interacting particles, forming a linear track and a rotor, and working as a molecular motor. Newton equations with damping for the rotor coordinate on the track x, with a prescribed time-dependence of the rotor angle Θ, indicated unidirectional motion of the rotor. Here, for the same system, the treatment was generalized to nonzero temperatures by including stochastic forces and treating both x and Θ via two coupled Langevin equations. Numerical results are reported for stochastic homogeneous distributions of impact events and Gaussian distributions of stochastic forces acting on both the variables. For specific values of parameters involved, the unidirectional motion of the rotor along the track is confirmed, but with a mechanism that is not necessarily the same as that one by Porto. In an additional weak homogeneous potential field U(x)=const.x acting against the motion, the unidirectional motion persists. Then the rotor accumulates potential energy at the cost of thermal stochastic forces from the bath. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

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Figure 1
Plot of the dimensionless rotor-track potential energy ψ for γ = +1 as a function of X and Θ. Full article ">Figure 2
Time dependence of angle Θ as a function of T for γ = +1 and the same values of all the input parameters as specified in the main text. In all three cases, the impact times as well as the random forces were identical. Only in time intervals between any two individual impacts, different integration time steps (1/32, 1/16, 1/10, 1/8 and 1/7 for curves a) to e), respectively) were used. The onset of chaos appears at about T = 1200. At T = 1000, the values of Θ obtained still coincided to 5 digits. Full article ">Figure 3
The same for X as a function of T for the same values of all the input parameters. At T = 1000, the values of X obtained still coincided to 6 digits. Notice that all the curves increase with increasing time. Full article ">Figure 4
Dependence of X on T for five different sequences of the random numbers involved for the ‘forward gear’ γ = +1. Full article ">Figure 5
The same for the ‘reverse gear’ γ = −1. The same sequences of random numbers as in <a href="#entropy-06-00057-f004" class="html-fig">Figure 4</a> were used. Full article ">Figure 6
Details of time-dependence of X(T) (γ = +1) for curve ‘c’ of <a href="#entropy-06-00057-f004" class="html-fig">Figure 4</a> before the first impact event in Θ occurs. Times T of the first six impacts in X were 4.4506, 5.0489, 5.9216, 9.7589, 9.9581, and 10.7401. The first impact in Θ appears at T = 216.8751. Worth noticing is how the impacts cause, because of the periodicity of ψ, sudden changes of X = x/b (on the vertical axis) by integers. Full article ">Figure 7
Dependence of X on T for five different sequences of the random numbers involved for the ‘forward gear’ γ = +1. All the input parameters as well as the random forces were identical as in <a href="#entropy-06-00057-f004" class="html-fig">Figure 4</a>, but with the linear potential c · X, c = 0.01 added to ψ(X, Θ). Full article ">

265 KiB

Open AccessArticle

The adiabatic piston: a perpetuum mobile in the mesoscopic realm

by Bruno Crosignani, Paolo Porto Di and Claudio Conti

Entropy 2004, 6(1), 50-56; https://doi.org/10.3390/e6010050 - 11 Mar 2004

Cited by 9 | Viewed by 8772

Abstract

A detailed analysis of the adiabatic-piston problem reveals, for a finely-tuned choice of the spatial dimensions of the system, peculiar dynamical features that challenge the statement that an isolated system necessarily reaches a time-independent equilibrium state. In particular, the piston behaves like a [...] Read more.

A detailed analysis of the adiabatic-piston problem reveals, for a finely-tuned choice of the spatial dimensions of the system, peculiar dynamical features that challenge the statement that an isolated system necessarily reaches a time-independent equilibrium state. In particular, the piston behaves like a perpetuum mobile, i.e., it never comes to a stop but keeps wandering, undergoing sizeable oscillations around the position corresponding to maximum entropy; this has remarkable implications on the entropy changes of a mesoscopic isolated system and on the limits of validity of the second law of thermodynamics in the mesoscopic realm. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

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189 KiB

Open AccessArticle

The Deep Physics Behind the Second Law: Information and Energy As Independent Forms of Bookkeeping

by Todd L. Duncan and Jack S. Semura

Entropy 2004, 6(1), 21-29; https://doi.org/10.3390/e6010021 - 11 Mar 2004

Cited by 16 | Viewed by 8170

Abstract

Even after over 150 years of discussion, the interpretation of the second law of thermodynamics continues to be a source of confusion and controversy in physics. This confusion has been accentuated by recent challenges to the second law and by the difficulty in [...] Read more.

Even after over 150 years of discussion, the interpretation of the second law of thermodynamics continues to be a source of confusion and controversy in physics. This confusion has been accentuated by recent challenges to the second law and by the difficulty in many cases of clarifying which formulation is threatened and how serious the implications of a successful challenge would be. To help bring clarity and consistency to the analysis of these challenges, the aim of this paper is to suggest a simple formulation of the deep physics of the second law, and to point out how such a statement might help us organize the challenges by level of seriousness. We pursue the notion that the second law is ultimately a restriction operating directly on the dynamics of information, so the existence of this law can be traced to the need for a system of "information bookkeeping" that is independent of the bookkeeping for energy. Energy and information are related but independent, so the dynamical restrictions for one cannot be derived from those for the other. From this perspective, we also suggest the possibility that the foundation of the second law may be linked to the finite capacity of nature to store bookkeeping information about its own state. Full article

(This article belongs to the Special Issue Quantum Limits to the Second Law of Thermodynamics)

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Entropy, Volume 6, Issue 1 (March 2004) – 20 articles , Pages 1-232

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