An electrothermal instability in a conducting wire: Homogeneous and inhomogeneous stationary states for an exactly solvable model

An exactly solvable model of the ballast resistor is considered. Analytic expressions are obtained for the nonuniform stationary temperature distributions and the correspondingI–V characteristics. A bifurcation point for Neumann boundary conditions is found and its analytic properties are discussed. It is found that the infinite wire limit plays a role analogous to the thermodynamic limit in statistical mechanics for equilibrium phase transitions.Dedicated to the memory of our colleague and friend Pierre Résibois.     

Minimal Brownian ratchet: an exactly solvable model

We develop an analytically solvable three-state discrete-time minimal Brownian ratchet (MBR), where the transition probabilities between states are asymmetric. By solving the master equations, we obtain the steady-state probabilities. Generally, the steady-state solution does not display detailed balance, giving rise to an induced directional motion in the MBR. For a reduced two-dimensional parameter space, we find the null curve on which the net current vanishes and detailed balance holds. A system on this curve is said to be balanced. On the null curve, an additional source of external random noise is introduced to show that a directional motion can be induced under the zero overall driving force.     

An exactly solvable model for the large polaron

We present an exactly diagonalizable model Hamiltonian for the large polaron derived by analyzing the variational ansatz by Haga-Larsen (HL) for the Fröhlich Hamiltonian. The lowest energy eigenvalue of the model Hamiltonian for fixed wave numbers reproduces the energy of the variational ansatz by Haga-Larsen and is, therefore, an upper bound with respect to the corresponding energy eigenvalue of the Fröhlich Hamiltonian. This is valid for any momentum which is proven by extending the Haga-Larsen approach. Furthermore, since all integrations can be performed analytically, the model Hamiltonian is easily tractable. The energy eigenvalue spectrum of the model Hamiltonian is studied below and above the phonon-emission threshold. The quality of the model Hamiltonian is determined by the variational ansatz of Haga and Larsen. Incorporating an improved energy-momentum relation, a generalized model Hamiltonian is derived possessing a larger validity range with respect to the coupling strength. Furthermore, a second exactly diagonalizable model Hamiltonian based on improved Wigner-Brillouin perturbation theory due to Warmenbol, Peeters, and Devreese (WPD) is presented. It is briefly demonstrated that one is able to construct all mentioned model Hamiltonians also in the 2D polaron problem. In contrast to the 3D case, where the HL-type model Hamiltonian possesses the higher quality for any momentum, in the 2D case, it works well only for small momenta. For large momenta, only the WPD-type model Hamiltonian describes the energy-momentum relation correctly. We demonstrate the usefulness of the model Hamiltonian concept by exactly calculating the one-electron Green’s function for all mentioned model Hamiltonians and comment why significant advantages of the model Hamilton concept for the treating of low-dimensional systems (planar semiconducting quantum-well structures) can be expected.     

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.     

An exactly solvable model of a multidimensional incommensurate structure

The paper considers the class of Schrödinger multidimensional discrete operators with quasi-periodic unbounded potential for which essentially complete spectral analysis may be carried out. In the case of sufficiently high incommensurability of almost-periods, the spectrum of such operators is found to be pure point and simple, the eigenfunctions exponentially localized and the low frequency conductivity exponentially small. In the one-dimensional case, for any incommensurability, the spectrum does not contain the absolutely continuous component, while for small incommensurability the spectrum is singular continuous.     

An exactly solvable model of the phonon-assisted hopping dc conductivity

The model, which is commonly used to obtain the Mott law for the temperature dependence of the dc conductivity of elemental covalent amorphous semiconductors, is solved exactly in the case of small electron-phonon coupling. Differences between the Mott law and our solution are discussed.     

Domain growth of first-order phase transitions: An exactly solvable model

We consider a simple exactly solvable model of an order-disorder transition: noninteracting interfaces moving with constant speed in one dimension. We obtain a linear increase with time of the average domain size and a weak oscillatory behavior of the scattering structure function.     

An exactly solvable linear model giving indication of stochastic resonance

The linear stochastic equation dx _β /dt+[1+f _β (t)]x _β (t)=A sin (Ωt) is discussed. The functionƒ _β (t) is defined as a Poissonian noise dependent on a parameterβ>0,ƒ _β (t)=β Σ _j [δ(t − t _j ⁺ ) −δ (t − t _j ⁻ )]. The mean frequency of the delta-pulses is chosen asβ-dependent in the formλ(β)=2γ(β ⁻² + 1) exp(−β) whereγ is a constant from the interval (0, 0.974). With the stochastic functionƒ _β (t) defined in this way, attention is paid on the oscillational term of the averaged function 〈x(t)〉, 〈x(t)〉_osc=Āsin(Ωt − α). It is found that the dependenceĀ=Ā(β) exhibits one maximum and one minimum. The occurrence of these extrema seems to affirm the presence of stochastic resonance. This work has been supported by the Slovak Grant Agency VEGA under contract No. 1/4319/97.     

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.

     

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.

     

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.     

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.     

Simulation calculation of dielectric constants: Comparison of methods on an exactly solvable model

A mean spherical model of classical dipoles on a simple cubic lattice of sideM=2N+1 sites is considered. Exact results are obtained for finite systems using periodic boundary conditions with an external dielectric constant and using reaction field boundary conditions with a cutoff radiusR _c N and an external dielectric constant. The dielectric constant in the disordered phase is calculated using a variety of fluctuation formulas commonly implemented in Monte Carlo and molecular dynamics simulations of dipolar systems. The coupling in the system is measured by the parametery=4 ²/9kT, where ² is the fixed mean square value of the dipole moments on the lattice. The system undergoes a phase transition aty2.8, so that very high dielectric constants cannot be obtained in the disordered phase. The results show clearly the effects of system size, cutoff radius, external dielectric constant, and different measuring techniques on a dielectric constant estimate. It is concluded that with periodic boundary conditions, the rate of approach of the dielectric constant estimate to its thermodynamic limit is asN ^–2/3 and depends only weakly on. Methods of implementing reaction field boundary conditions to give rapid convergence to the thermodynamic limit are discussed.     

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.