Ehrenfest Equation Research Articles

Flutter of a beam in supersonic flow: truncated version of Timoshenko–Ehrenfest equation is sufficient

This paper deals with flutter due to gas flow of a uniform and homogeneous beam with shear deformation and rotary inertia effects taken into account. It utilizes the truncated, consistent Timoshenko–Ehrenfest beam model in contrast to the original Timoshenko–Ehrenfest equations. Omission of the fourth order derivative term in the governing differential equation of the original equations of Timoshenko and Ehrenfest is shown to be more consistent than retaining it as is done in the original Timoshenko–Ehrenfest set. The original Timoshenko–Ehrenfest equations are exposed in almost all textbooks published after 1921. However, this paper shows that these original equations are unnecessarily overcomplicated, and simpler set is consistent. This truncation significantly simplifies the analytical and numerical analyses considerably and agrees with the principle of parsimony, i.e. the most acceptable explanation of a phenomenon is the simplest, involving the fewest entities and assumptions. The critical flutter velocities are compared with the results obtained by using the original set of Timoshenko–Ehrenfest equations.

Three alternative versions of the theory for a Timoshenko–Ehrenfest beam on a Winkler–Pasternak foundation

This paper analyzes the free vibration frequencies of a beam on a Winkler–Pasternak foundation via the original Timoshenko–Ehrenfest theory, a truncated version of the Timoshenko–Ehrenfest equation, and a new model based on slope inertia. We give a detailed comparison between the three models in the context of six different sets of boundary conditions. In particular, we analyze the most common combinations of boundary conditions deriving from three typical end constraints, namely the simply supported end, clamped end, and free end. An interesting intermingling phenomenon is presented for a simply-supported (S-S) beam together with proof of the ‘non-existence’ of zero frequencies for free-free (F-F) and simply supported-free (S-F) beams on a Winkler–Pasternak foundation.

High-pressure dielectric studies on 1,6-anhydro-β-D-mannopyranose (plastic crystal) and 2,3,4-tri-O-acetyl-1,6-anhydro-β-D-glucopyranose (canonical glass).

Broadband Dielectric Spectroscopy was applied to investigate molecular dynamics of two anhydrosaccharides, i.e., 1,6-anhydro-β-D-mannopyranose, anhMAN (hydrogen-bonded system) and 2,3,4-tri-O-acetyl-1,6-anhydro-β-D-glucopyranose, ac-anhGLU (van der Waals material), at different thermodynamic conditions. Moreover, the reported data were compared with those recently published for two other H-bonded systems, i.e., 1,6-anhydro-β-D-glucopyranose (anhGLU) and D-glucose (D-GLU). A direct comparison of the dynamical behavior of the materials with a similar chemical structure but significantly differing by the degrees of freedom, complexity, and intermolecular interactions made it possible to probe the impact of compression on the fragility, Temperature-Pressure Superpositioning and pressure coefficient of the glassy crystal/glass transition temperatures (dTgc/dp ; dTg/dp). Moreover, the correlation between dTgc/dp determined experimentally from the high-pressure dielectric data and the Ehrenfest equation has been tested for the plastic crystals (anhGLU and anhMAN) for the first time. Interestingly, a satisfactory agreement was found between both approaches. It is a quite intriguing finding which can be rationalized by the fact that the studied materials are characterized by the low complexity (lower degrees of freedom with respect to the molecular mobility) as well as ordered internal structure. Therefore, one can speculate that in contrast to the ordinary glasses the dynamics of the plastic crystals might be described with the use of a single order parameter. However, to confirm this thesis further, pressure-volume-temperature (PVT) experiments enabling calculations of the Prigogine Defay ratio are required.

‘Second’ Ehrenfest equation for second order phase transition under hydrostatic pressure

It is shown that the fundamental conditions for the second-order phase transitions and , from which the two Ehrenfest equations follow (the ‘usual’ and the ‘second’ ones), are realised only at zero hydrostatic pressure (). At the volume jump ΔV at the transition is proportional to the pressure and to the jump of the compressibility ΔζV, whereas the entropy jump ΔS is proportional to the pressure and to the jump of the thermal expansion coefficient ΔαV. This means that at non-zero hydrostatic pressure the phase transition is of the first order and is described by the Clausius–Clapeyron equation. At small pressure this equation coincides with the ‘second’ Ehrenfest equation . At high P, the Clausius–Clapeyron equation describes qualitatively the caused by the crystal compression positive curvature of the dependence.

Density power law and structures of metallic glasses

The existence of a universal power law relating the position of the first sharp diffraction peak (q, FSDP) to the density (ρ or the volume V) with a constant exponent <3 has been debated in the last decade. A constant dimensionality is important because it reflects the fractal topology of the glass structures. In this study, the validity of the Ehrenfest equation applied to multi-component metallic glasses is examined using first-principles molecular dynamics calculations. The results show that the Ehrenfest coefficient depends on the local structures of the glasses and is not a constant for all glasses. Moreover, since the diffraction pattern is determined by the scattering between atom pairs, in a multi-component glass, the X-ray diffraction FSDP is only sensitive to the heavy atoms, and the observed P-q relationship does not necessary correspond to the PV equation of state of the bulk material and is not always a suitable indicator for monitoring structural phase transitions or volume changes. On the other hand, for suitable systems, neutron diffraction is a reliable method to determine the structural features of both heavy and light atoms. In this study, the simulated neutron diffraction patterns of Ca72·7Al27.3 metallic glasses show a clear splitting of the FSDP at the pressure where the pressure-induced polyamorphism transition occurs. From the presented results, there is no justification for expecting the existence of a universal power scaling law with a constant exponent for all glasses.

About the proof that the s-wave functions of the hydrogen atom obey the Ehrenfest equation of motion

Applying the concept of quantum hydrodynamics on a hydrogen atom, one can derive an equation of motion which describes the time derivative of the mass current density for the relative motion of the electron and the proton. Since this derivation can be performed with the Ehrenfest theorem, we call this equation the Ehrenfest equation of motion. As quantum hydrodynamics is consistent with Schrödinger’s quantum mechanics, the stationary s-wave functions satisfy both the Schrödinger equation and the Ehrenfest equation of motion. It is the purpose of this paper to prove that the stationary s-wave functions satisfy the Ehrenfest equation of motion. From the quantum hydrodynamical point of view, this result for s-wave functions can be interpreted that the Coulomb force density that attracts the electron to the proton is compensated by a quantum force density that is related to the dispersion of the probability density.

Geometric Phase Effects in Nonadiabatic Dynamics near Conical Intersections

Dynamical consideration that goes beyond the common Born-Oppenheimer approximation (BOA) becomes necessary when energy differences between electronic potential energy surfaces become small or vanish. One of the typical scenarios of the BOA breakdown in molecules beyond diatomics is a conical intersection (CI) of electronic potential energy surfaces. CIs provide an efficient mechanism for radiationless electronic transitions: acting as "funnels" for the nuclear wave function, they enable rapid conversion of the excessive electronic energy into the nuclear motion. In addition, CIs introduce nontrivial geometric phases (GPs) for both electronic and nuclear wave functions. These phases manifest themselves in change of the wave function signs if one considers an evolution of the system around the CI. This sign change is independent of the shape of the encircling contour and thus has a topological character. How these extra phases affect nonadiabatic dynamics is the main question that is addressed in this Account. We start by considering the simplest model providing the CI topology: two-dimensional two-state linear vibronic coupling model. Selecting this model instead of a real molecule has the advantage that various dynamical regimes can be easily modeled in the model by varying parameters, whereas any fixed molecule provides the system specific behavior that may not be very illustrative. After demonstrating when GP effects are important and how they modify the dynamics for two sets of initial conditions (starting from the ground and excited electronic states), we give examples of molecular systems where the described GP effects are crucial for adequate description of nonadiabatic dynamics. Interestingly, although the GP has a topological character, the extent to which accounting for GPs affect nuclear dynamics profoundly depends on topography of potential energy surfaces. Understanding an extent of changes introduced by the GP in chemical dynamics poses a problem of capturing GP effects by approximate methods of simulating nonadiabatic dynamics that can go beyond simple models. We assess the performance of both fully quantum (wave packet dynamics) and quantum-classical (surface-hopping, Ehrenfest, and quantum-classical Liouville equation) approaches in various cases where GP effects are important. It has been identified that the key to success in approximate methods is a method organization that prevents the quantum nuclear kinetic energy operator to act directly on adiabatic electronic wave functions.

Computer-aided cluster expansion: An efficient algebraic approach for open quantum many-particle systems

We introduce an equation of motion approach that allows for an approximate evaluation of the time evolution of a quantum system, where the algebraic work to derive the equations of motion is done by the computer. The introduced procedures offer a variety of different types of approximations applicable for finite systems with strong coupling as well as for arbitrary large systems where augmented mean-field theories like the cluster expansion can be applied. Program summaryProgram Title: EoM_main.frmProgram Files doi:http://dx.doi.org/10.17632/fjwxr28j3d.1Licensing provisions: CC By 4.0Programming language:FORMNature of problem: Quantum many-particle systems are an important subject in fundamental and applied research. The calculation of the time evolution of such systems is a key aspect to investigate and understand their properties. In most cases the Hilbert space that represents the quantum mechanical system is too big to be processed by numerically exact methods and approximation methods have to be used.Solution method: The program automates an equation-of-motion technique that uses the generalized Ehrenfest equation to derive the time evolution for expectation values of physical observables. The cluster expansion is used to close the hierarchy of the equations of motion. The offered method allows for a variety of different types of approximations to solve such problems with small numerical effort [1]. [1] Leymann, H.A.M., Foerster, A., Wiersig, J., 2014. Expectation value based equation-of-motion approach for open quantum systems: A general formalism. Phys. Rev. B 89 (8), 085308.

High pressure studies of the phase transition in the ferroelectric Sn2P2S6

Abstract We apply a method of pulse-adiabatic modulation of pressure to obtain heat capacity and thermal expansion of ferroelectric Sn2P2S6 in the vicinity of the second order phase transition at pressures to 5 kbar. The phase transition in Sn2P2S6 does not change its nature and stays second order in the whole range of pressure currently studied. The earlier conclusion on the tricritical features of the phase transition in Sn2P2S6 cannot be confirmed. Discontinuities of heat capacity and thermal expansion perfectly fit the Ehrenfest equation that expected in the mean field theories. An excellent performance of the Ehrenfest formula in a wide range of pressures establishes phase transition in Sn2P2S6 as an almost ideal mean field phase transition.

Ehrenfest equations for second-order phase transition under hydrostatic and uniaxial stresses

ABSTRACTFor 40 years, many research works have been confusing the linear part of the Ehrenfest equation for the shift of the second-order phase transition temperature in solids by the hydrostatic pressure with the equation for the shift caused by the uniaxial pressure . While the sum of the three slopes is trivially equal to the slope due to hydrostatic pressure , the sum of calculated from the exact Ehrenfest equation for the uniaxial pressure generally does not coincide with and may differ from by the sign and by the absolute value by more than three times.

Extended Ehrenfest theorem with radiative corrections

A set of basic evolution equations for the mean values of dynamical variables is obtained from the Fokker–Planck equation applied to the general problem of a particle subject to a random force. The specific case of stochastic electrodynamics is then considered, in which the random force is due to the zero-point radiation field. Elsewhere it has been shown that when this system reaches a state of energy balance, it becomes controlled by an equation identical to Schrödinger’s, if the radiationless approximation is made. The Fokker–Planck equation was shown to lead to the Ehrenfest theorem under such an approximation. Here we show that when the radiative terms are not neglected, an extended form of the Ehrenfest equation is obtained, from which follow, among others, the correct formulas for the atomic lifetimes and the (nonrelativistic) Lamb shift.

Equivalent linearization technique for quantum anharmonic oscillators

Quantum dynamics means studying the evolution of an initially prescribed wave function. This is analytically tractable for special wavefunctions for the simplest of the situations—free particle and simple harmonic oscillator. The purely anharmonic oscillators are virtually impossible to handle. We show here that the study of Ehrenfest's equation provides an alternative route to studying quantum dynamics. It does not give exact answers but clarifies some basic aspects of quantum dynamics by providing a prescription for constructing equivalent simple harmonic oscillators.

Description of Phase Transition in a Black Hole with Conformal Anomaly in the Ehrenfest’s Scheme

We make use of the Ehrenfest's equation to explore the phase transition of a black hole with conformal anomaly. The first order phase transition is ruled out because no discontinuity appears in entropy of the black holes. We find that the phase transition of black holes belong to the second order subject to the Ehrenfest's equations. Further we also show that the second order phase transition will not happen for the black holes without conformal anomaly.

Second-order phase transitions and Ehrenfest equation for strained solids

The equations describing the second-order phase transitions at the hydrostatic and non-hydrostatic pressures are considered. It is shown that the proportionality coefficient between an “effective” volume and the true one V∗ = AV is inversely proportional to the compressibility of the solid at a uniaxial pressure and has a jump at the second-order phase transition. In the case of the non-hydrostatic pressure the “effective” volume of the solid is not a continuous function of temperature and has a jump at the phase transition as well. The Ehrenfest equation is generalized to the solids with an arbitrary homogeneous elastic deformation accompanied by change of the solid volume, in particular, to the solid strained by the uniaxial, biaxial or triaxial pressure. It is shown that the sum of the derivatives of the phase transition temperature with respect to uniaxial pressures applied along axes a, b, c does not coincide with the derivative of the phase transition temperature with respect to the hydrostatic pressure.

New type of phase transition in Reissner Nordström–AdS black hole and its thermodynamic geometry

The phase transition of a RN–AdS black hole is studied in details using Ehrenfest's equations. There is no discontinuity in entropy which signals a lack of any first order phase transition. We then show that although Ehrenfest's first equation is satisfied, the second is not, so that a true second order phase transition is also ruled out. However this deviation from the second Ehrenfest's equation, for a certain range of the black hole charge, indicates the existence of a new glassy type transition. We finally study the thermodynamic behaviour using state space geometry and find that the scalar curvature diverges exactly at those points where the heat capacity diverges.

Thermodynamical relations at characteristic points

We define the characteristic temperature T C of a material by using the behaviour of the specific heat in the vicinity of this temperature: C p,N (T ) = f(1 − (T/T C ),p,N ) for T < T C and C p,N (T ) = g((T/T C ) − 1,p,N ) for T > T C where p is the external pressure and N is the number of particles. The characteristic temperature T C = T C ( p, N ) can e.g. be identified with the transition temperature in the case of continuous phase transitions or with one of the two characteristic temperatures (on heating) and (on cooling) in the case of discontinuous phase transitions. Using the idea of the characteristic temperature, we present a new method on how to calculate the Gibbs potential G = G(T, p, N) in the vicinity of the characteristic temperature, without it being a necessity to know the explicite expressions for f and g. This general and uniform approach allows to derive many thermodynamical relations connecting measurable quantities at the characterisic point as e.g. Clausius–Clapeyron equation and its analogs, valid for discontinuous phase transitions and 15 relations valid for both continuous and discontinuous phase transitions, 6 of them are new. As a special case of these latter relations, we find six relations valid for continuous phase transitions (Ehrenfest equation and its analogs). The derived relations result exclusively from the definition of the characteristic point and are independent from the underlying microscopical mechanism leading to phase transitions (as e.g. solid–liquid, liquid–vapour, magnetic, ferroelectric, structural and superconducting transitions) and therefore, they are generally valid. These relations can be very important for material science. Using some of them it is possible to predict (without performing additional experiments) whether the transition or characteristic temperature T C of a given material sample will increase or decrease under doping (injection) or applied pressure. Similar 24 thermodynamical relations can also be found with ease for systems described by the Helmholtz potential F = F(T,V, N ).

The effect of mechanical strains on the ferroelectric and dielectric properties of a model single crystal – Phase field simulation

The effect of applied mechanical strains on the ferroelectric and dielectric properties of a model single crystal is investigated using a phase field model based on the time-dependent Ginzburg–Landau equation, which takes both multiple-dipole–dipole-electric and -elastic interactions into account. The evolution of the ferroelectric domain structure is simulated at different temperatures and applied strains. The results show that the paraelectric/ferroelectric phase transition temperature linearly increases with the applied mechanical strain under mechanical clamping conditions. Analogous to the classical Ehrenfest equation, a thermodynamics equation is derived to describe the relationship between the transition temperature and the applied strain. The change in the domain structure with temperature under applied inequiaxial strains is different from that under applied equiaxial strains. The simulations also illustrate the changes in the coercive field, the remanent polarization and the nonlinear dielectric constant with the applied strain.

Effect of Hydrostatic Pressure on P-14M-2M and P-2M Martensitic Transformations in Single Crystalline Ni&amp;ndash;Mn&amp;ndash;Ga Ferromagnetic Shape Memory Alloys

Effect of hydrostatic pressure on martensitic transformation temperature has been investigated in single crystalline Ni 2 . 1 4 Mn 0 . 8 4 Ga 1 . 0 2 and Ni 2 . 1 4 Mn 0 . 9 2 Ga 0 . 9 4 ferromagnetic shape memory alloys, which exhibit P- 14M-2M and P-2M transformations, respectively, where P stands for a parent phase. We found that all the martensitic transformation temperatures increase linearly with increasing hydrostatic pressure; their increasing rates against hydrostatic pressure are 3 K/GPa (P → 14M), 13 K/GPa (14M → 2M) and 7 K/GPa (P → 2M). The volume changes associated with the transformations are calculated to be -0.04% (P → 14M), -0.04% (14M → 2M) and -0.15% (P → 2M) by using the Clausius-Clapeyron equation. In addition, Curie temperature of the parent phase of Ni 2 . 1 4 Mn 0 . 8 4 Ga 1 . 0 2 increases linearly with increasing hydrostatic pressure; its increasing rate of about 4 K/GPa is well explained by using the Ehrenfest equation.

High‐pressure rheology of eight synthetic lubricants based on phase diagrams

AbstractNew sets of equations have been constructed from the density‐pressure‐temperature and viscosity‐pressure‐temperature relationships to determine the rheology of synthetic lubricants at high pressure. A phase diagram was first plotted for the oils up to a pressure of 1.2 GPa and temperatures from 303 to 353 K using high‐pressure density experimental results and others. For the oils at temperatures of 313, 333, and 353 K, the high‐pressure viscosity experiment was done up to a viscosity of 104 Pa s and up to a pressure of 0.4 GPa. A relationship, based on free volume theory, between pressure and temperature with varying density and viscosity was proposed. It was found that the new equations from these relationships could predict the high‐pressure rheology more closely than those of previous researchers. Moreover, it was shown that the Ehrenfest equation or the gradient of the phase diagram is closely related to the expansion coefficient of free volume.

Chaos in the quantum double well oscillator: the Ehrenfest view revisited

We treat the double well quantum oscillator from the standpoint of the Ehrenfest equation but in a manner different from Pattanayak and Schieve. We show that for short times there can be chaotic motion due to quantum fluctuations, but over sufficiently long time there will be quantum noise induced oscillations between the two wells, thus providing an alternative approach to the quantum noise induced chaotic oscillations found recently by Bag and Ray.