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.     

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.     

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.

     

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.     

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.     

Interpretation of an exactly solvable two-component plasma

The interpretation of the exact calculation of the partition function and correlations of a two-component plasma obtained earlier is considered. The system has species of charge ratio 12 which are constrained to lie on a circle and interact via the two-dimensional Coulomb potential. By studying the exact results we gain an understanding of why the excess thermodynamic quantities of the two component system can be well approximated by the sum of the appropriate excess thermodynamic quantities of the one-component systems.     

An exactly solvable model for two-dimensional non-Hermitian quasicrystals

The phenomenon Anderson localization explains the metalinsulator transition in a material with the increase of disorder and its electrons’transport change from diffusive into localized.The study of the Anderson localization has been extended to many fields of physics,including the quasiperiodic or incommensurate systems.     

An exactly solvable infinite-ranged model for sliding charge-density-waves

An infinite-ranged model for the sliding charge-density-wave (CDW) with periodic harmonic impurity potentials is shown to be solvable exactly in a whole range of the electric field strength. The solution yields some distinct features of the sliding CDW which were not reported in the same infinite-ranged model but with a sinusoidal impurity potential. The results on the nonlinear conductivity are discussed in comparison with the experiments on NbSe₃.