Feynman's ratchet and pawl: an exactly solvable model

We introduce a simple, discrete model of Feynman's ratchet and pawl, operating between two heat reservoirs. We solve exactly for the steady-state directed motion and heat flows produced, first in the absence and then in the presence of an external load. We show that the model can act both as a heat engine and as a refrigerator. We finally investigate the behavior of the system near equilibrium, and use our model to confirm general predictions based on linear-response theory.     

Heat flow in an exactly solvable model

A chain of one-dimensional oscillators is considered. They are mechanically uncoupled and interact via a stochastic process which redistributes the energy between nearest neighbors. The total energy is kept constant except for the interactions of the extremal oscillators with reservoirs at different temperatures. The stationary measures are obtained when the chain is finite; the thermodynamic limit is then considered, approach to the Gibbs distribution is proven, and a linear temperature profile is obtained.     

Correlation functions in an exactly solvable tefface-ledge-kink model

We solve exactly a terrace-ledge-kink (TLK) model describing a vicinal section of a crystal surface at a microscopic level, with either repulsive or attractive interactions between the ledges. As expected there is a faceting, or reconstructive, phase transition, driven either by temperature or by the chemical potential, that controls the mean slope of the surface. In the rough phase we carry out a thorough investigation of microscopic thermal fluctuations of the interface. This is done by combining Bethe ansatz and Conformal Field Theory methods in order to calculate appropriately defined correlators.     

An exactly solvable impurity model

We present an integrable model describing the interaction of conduction electrons with an arbitrary number of spin-1/2 impurities in one-dimension. The solution of the model is obtained by means of the Bethe function in the case when the number of particles is not conserved. The impurity magnetization, the Kondo-limit, the specific-heat coefficient are calculated on the basis of the Bethe ansatz solution.     

The reduced α-amplitude in an exactly solvable model

In recent work by Fliessbach the removal of an α-particle from a nucleus under the influence of a perturbation was considered. Using certain approximations the many-body transition matrix element was reduced to a one-body matrix element. This one-body matrix element showed that the appropriate bound α-amplitude in the initial nucleus (reduced amplitude) depends on the energy transferred to the removed α-particle. The present paper deals with an analytic model in which the one-body transition matrix element as given in that work can be derived exactly from the original microscopic matrix element.     

An exactly solvable two-dimensional biomembrane model

A model for a two-dimensional lipid bilayer, in which both short-range repulsive forces and long-range attractive forces piay a role, is solved exactly. First, in the absence of attractive forces, the configurational entropy is calculated by restricting the lipid hydrocarbon tails to a lattice and solving the resulting counting problem exactly. When long-range attractive forces which decrease exponentially with distance are acting between the polar head groups of the lipids, the partition function can still be calculated exactly, using integration over Gaussian random fields. The thermodynamic functions show a singularity which reflects the completion of the process of self-assemblage of the lipid bilayer. Finally, the model is used to test an approximation method for lipid bilayers.     

Topological order in an exactly solvable 3D spin model

We study a 3D generalization of the toric code model introduced recently by Chamon. This is an exactly solvable spin model with six-qubit nearest-neighbor interactions on an FCC lattice whose ground space exhibits topological quantum order. The elementary excitations of this model which we call monopoles can be geometrically described as the corners of rectangular-shaped membranes. We prove that the creation of an isolated monopole separated from other monopoles by a distance R requires an operator acting on Ω(R²) qubits. Composite particles that consist of two monopoles (dipoles) and four monopoles (quadrupoles) can be described as end-points of strings. The peculiar feature of the model is that dipole-type strings are rigid, that is, such strings must be aligned with face-diagonals of the lattice. For periodic boundary conditions the ground space can encode 4g qubits where g is the greatest common divisor of the lattice dimensions. We describe a complete set of logical operators acting on the encoded qubits in terms of closed strings and closed membranes.     

Distribution of pore-size in an exactly solvable two-dimensional biomembrane model

A model for a two-dimensional lipid bilayer in which both short range repulsive forces and long range attractive forces play a role, and which can be solved exactly, is discussed. It is shown that the bilayer consists of long stretches of relatively densely packed lipids separated by small pores. The statistical distributions of number and size of the pores are calculated from first principles.     

Static polarizability of an exactly solvable model of diatomic molecule

Summary A very simple one-dimensional model for a one-electron diatomic molecule, under the influence of a weak static electric field, is investigated by perturbation theory. The dipole polarizabilities for the two only bound states supported by this molecular model are evaluated as a function of the internuclear distance.

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Riassunto Si studia perturbativamente l'influenza di un debole campo elettrico statico su un modello monodimensionale semplificato di molecola biatomica contenente un unico elettrone “attivo”. Si calcola la polarizzabilità di dipolo nei due soli stati legati supportati dal modello in questione, mettendo in evidenza la dipendenza dalla distanza internucleare.

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Резюме В рамках теории возмущений исследуется влияние слабого статического злектрического поля на улрощенную одномерную модель двухатомной молекулы с одним “активным” злектроном. Вычисляется дилольная поляризуемость для двух связанных состояний, долускаемых рассматриваемой моделью молекулы, как функция расстояния между ядрами.

     

Fractional statistics in one dimension: view from an exactly solvable model

One-dimensional fractional statistics is studied using the Calogero-Sutherland model (CSM) which describes a system of non-relativistic quantum particles interacting with an inverse-square two-body potential on a ring. The inverse-square exchange can be regarded as a pure statistical interaction and this system can be mapped to an ideal gas obeying the fractional exclusion and exchange statistics. The details of the exact calculations of the dynamical correlation functions or this ideal system is presented in this paper. An effective low-energy one-dimensional “anyon” model is constructed; and its correlation functions are found to be in agreement with those in the CSM; and this agreement provides an evidence for the equivalence of the first- and the second-quantized construction of the 1D anyon model at least in the long-wavelength limit. Furthermore, the finite-size scaling applicable to the conformally invariant systems is used to obtain the complete set of correlation exponents for the CSM.     

Fluctuation effects in an exactly solvable model of phase transitions

An investigation is made of an exactly solvable phase transition model, which takes account of interaction only between fluctuations with equal and antiparallel momenta. It is shown that within this model fluctuation-induced first-order phase transitions are possible in agreement with renormalization group analysis predictions. The model is generalized phenomenologically, which leads to correct values of all large critical exponents. The effect of frozen-in impurities on a phase transition is studied and it is shown that in a narrow range proportional to the impurity concentration the phase transition smears.     

Dynamic polarizability of an exactly solvable model of diatomic molecule

Summary A very simple one-dimensional model for a one-electron diatomic molecule under the agency of a homogeneous, time-dependent electric field is investigated in the linear-response approximation. The behaviour of the dynamic polarizability is studied as a function of the frequency ω and the internuclear separationR.

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Riassunto Nell'approssimazione della risposta lineare si studia l'influenza di un (debole) campo elettrico dipendente dal tempo su un semplice modello monodimensionale di molecola biatomica omonucleare contenente un unico elettrone ?attivo?. Si studia la dipendenza della polarizzabilità dinamica dalla frequenza ω e dalla distanza internucleareR.

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Резюме В приближении линейного отклика рассматривается очень простая одномерная модель для одноэлектронной двухатомной молекулы при наличии (слабого) электрического поля, зависящего от времени. Исследуется поведение динамической поляризуемости, как функции частоты ω и расстояния между ядрамиR.

     

Negative-parity bands and gapless superconductivity in an exactly solvable model

In addition to the positive-parity yrast states discussed previously, exact solutions of the R(8) (two-level) model are found, interpretable as negative-parity yrast states. Similar solutions for the three- and four-level models are also considered. These are of special interest in connection with the phenomenon of gapless superconductivity. The states in question usually originate from two-quasiparticle rotational bands, but for some values of the spin, four- and six-quasi-particle bands may become “yrast.” With exact solutions available, the self-consistent cranking approximation is tested on these excited bands.     

An exactly solvable 5-state IRF model

A 5-state IRF model (σ_l = 0, 1, 2, 3, 4) with a hard-core condition 0 ⩽ σ_i + σ_j ⩽ 4 on adjacent spins σ_i and σ_j is exactly solved. Explicit parametrizations of 70 independent Boltzmann weights which satisfy 510 star-triangle equations are presented with an exact expression for the free energy.