Microcanonical Theory Research Articles

Geometric microcanonical theory of two-dimensional truncated Euler flows.

This paper presents a geometric microcanonical ensemble perspective on two-dimensional truncated Euler flows, which contain a finite number of (Fourier) modes and conserve energy and enstrophy. We explicitly perform phase space volume integrals over shells of constant energy and enstrophy. Two applications are considered. In the first part, we determine the average energy spectrum for highly condensed flow configurations and show that the result is consistent with Kraichnan's canonical ensemble description, despite the fact that no thermodynamic limit is invoked. In the second part, we compute the probability density for the largest-scale mode of a free-slip flow in a square, which displays reversals. We test the results against numerical simulations of a minimal model and find excellent agreement with the microcanonical theory, unlike the canonical theory, which fails to describe the bimodal statistics. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Improved microcanonical instanton theory.

Canonical (thermal) instanton theory is now routinely applicable to complex gas-phase reactions and allows for the accurate description of tunnelling in highly non-separable systems. Microcanonical instanton theory is by contrast far less well established. Here, we demonstrate that the best established microcanonical theory [S. Chapman, B. C. Garrett and W. H. Miller, J. Chem. Phys., 1975, 63, 2710-2716], fails to accurately describe the deep-tunnelling regime for systems where the frequencies of the orthogonal modes change rapidly along the instanton path. By taking a first principles approach to the derivation of microcanonical instanton theory, we obtain an improved method, which accurately recovers the thermal instanton rate when integrated over energy. The resulting theory also correctly recovers the separable limit and can be thought of as an instanton generalisation of Rice-Ramsperger-Kassel-Marcus (RRKM) theory. When combined with the density-of-states approach [W. Fang, P. Winter and J. O. Richardson, J. Chem. Theory Comput., 2021, 17, 40-55], this new method can be straightforwardly applied to real molecular systems.

Theory of the vortex-clustering transition in a confined two-dimensional quantum fluid

Clustering of like-sign vortices in a planar bounded domain is known to occur at negative temperature, a phenomenon that Onsager demonstrated to be a consequence of bounded phase space. In a confined superfluid, quantized vortices can support such an ordered phase, provided they evolve as an almost isolated subsystem containing sufficient energy. A detailed theoretical understanding of the statistical mechanics of such states thus requires a microcanonical approach. Here we develop an analytical theory of the vortex clustering transition in a neutral system of quantum vortices confined to a two-dimensional disk geometry, within the microcanonical ensemble. As the system energy increases above a critical value, the system develops global order via the emergence of a macroscopic dipole structure from the homogeneous phase of vortices, spontaneously breaking the Z2 symmetry associated with invariance under vortex circulation exchange, and the rotational SO(2) symmetry due to the disk geometry. The dipole structure emerges characterized by the continuous growth of the macroscopic dipole moment which serves as a global order parameter, resembling a continuous phase transition. The critical temperature of the transition, and the critical exponent associated with the dipole moment, are obtained exactly within mean-field theory. The clustering transition is shown to be distinct from the final state reached at high energy, known as supercondensation. The dipole moment develops via two macroscopic vortex clusters and the cluster locations are found analytically, both near the clustering transition and in the supercondensation limit. The microcanonical theory shows excellent agreement with Monte Carlo simulations, and signatures of the transition are apparent even for a modest system of 100 vortices, accessible in current Bose-Einstein condensate experiments.

Microcanonical unimolecular rate theory at surfaces. II. Vibrational state resolved dissociative chemisorption of methane on Ni(100).

A three-parameter microcanonical theory of gas-surface reactivity is used to investigate the dissociative chemisorption of methane impinging on a Ni(100) surface. Assuming an apparent threshold energy for dissociative chemisorption of E(0)=65 kJ/mol, contributions to the dissociative sticking coefficient from individual methane vibrational states are calculated: (i) as a function of molecular translational energy to model nonequilibrium molecular beam experiments and (ii) as a function of temperature to model thermal equilibrium mbar pressure bulb experiments. Under fairly typical molecular beam conditions (e.g., E(t)>/=25 kJ mol(-1), T(s)>/=475 K, T(n)</=400 K), sticking from methane in the ground vibrational state dominates the overall sticking. In contrast, under thermal equilibrium conditions at temperatures T>/=100 K the dissociative sticking is dominated by methane in vibrationally excited states, particularly those involving excitation of the nu(4) bending mode. Fractional energy uptakes f(j) defined as the fraction of the mean energy of the reacting gas-surface collision complexes that derives from specific degrees of freedom of the reactants (i.e., molecular translation, rotation, vibration, and surface) are calculated for thermal dissociative chemisorption. At 500 K, the fractional energy uptakes are calculated to be f(t)=14%, f(r)=21%, f(v)=40%, and f(s)=25%. Over the temperature range from 500 K to 1500 K relevant to thermal catalysis, the incident gas-phase molecules supply the preponderance of energy used to surmount the barrier to dissociative chemisorption, f(g)=f(t)+f(r)+f(v) approximately 75%, with the highest energy uptake always coming from the molecular vibrational degrees of freedom. The predictions of the statistical, mode-nonspecific microcanonical theory are compared to those of other dynamical theories and to recent experimental data.

Dissociative chemisorption of methane on Ni(100): Threshold energy from CH4(2ν3) eigenstate-resolved sticking measurements

A three-parameter microcanonical theory of gas-surface reactivity is used to model the dissociative sticking of vibrationally excited methane with two quanta of energy in the ν3 antisymmetric C–H stretch. An apparent threshold energy for C–H bond cleavage of CH4 incident on Ni(100) of 65 kJ/mol is found, in quantitative agreement with ab initio quantum chemistry calculations but 38 kJ/mol less than GGA-DFT calculations. Successful microcanonical analysis and prediction of recent thermal equilibrium and various nonequilibrium dissociative chemisorption experiments for methane on Ni(100) provide no evidence for mode-specific reactivity.

Assessing a microcanonical theory of gas-surface reactivity: Applicability to thermal equilibrium, nonequilibrium, and eigenstate-resolved dissociation of methane on Ni(100)

A simple, three-parameter microcanonical theory of gas-surface reactivity is shown to predict experimental dissociative sticking probabilities for methane dissociative chemisorption on the Ni(100) surface over roughly ten orders of magnitude variation in both pressure and sticking—even at quantum state resolved levels of detail. Facile energy randomization within the transiently formed gas-surface collision complexes is postulated to make the pooled energy from 15 local degrees of freedom statistically available to surmount the barrier to dissociation. The apparent threshold energy for C–H bond cleavage of CH4 incident on Ni(100) is 67 kJ/mol, down from 432 kJ/mol in the gas phase.

Towards a microscopic theory of nonisothermal stochastic processes

Using a nonlinear projection operator technique, the Markovian stochastic dynamics of a classical Brownian particle is derived in the case that the effective temperature is not constant. After recapitulating the general formalism and its application to isothermal Brownian motion, an isolated ‘particle plus environment’ system is analyzed in the microcanomical ensemble. The ensuing generalized Kramers equation is shown to involve a microcanonical potential of mean force different from the conditional free energy, and the conditional entropy emerges as the relevant equilibrium potential. The Smoluchowski limit is presented, and the effective spatial diffusion coefficient is calculated. Upon coupling the system to a heat bath, the microcanonical theory is seen to correspond to the case of small heat conductance. The final result for the stochastic process in the position, momentum, and energy variables is shown to be consistent with both isothermal Kramers theory and nonlinear equilibrium thermodynamics.

Dynamics of persistent collision complexes in molecular beam reactive scattering

Abstract New microcanonical theories for the angular distributions of reactive scattering arising from long-lived collision complexes and their application to a wide range of transition state geometries are reviewed. Transition states formed at the top of centrifugal barriers on potential energy surfaces without a potential energy barrier are well described by symmetric top transition states, provided that the loosening of bending mode vibrations into product rotation is included. A phase space description appropriate for collisions at small impact parameters can be incorporated into a consistent mathematical description. Tight transition states formed at the top of a potential energy barrier and involving H atom displacement are described by an asymmetrical top transition state theory which includes all the symmetric top configurations as limiting cases. However the effects of transition state flexibility rapidly obscure the observable consequences of preferred geometry for directions of dissociation awa...

Transition state dynamics of F + ICl, C2H5I reactive scattering

Reactive scattering of F atoms with ICl and C2H5I molecules has been studied at an initial translational energy E ∼ 12 kJ mol-1 using a supersonic beam of F atoms seeded in Ne buffer gas. The centre-of-mass angular distribution of IF scattering for F + ICl, measured previously at E ∼ 34 kJ mol-1, shows peaking mainly in the forward direction, which is consistent with reaction via a short-lived complex with a lifetime of half a rotational period. The microcanonical theory for dissociation of a persistent complex via a prolate symmetric top transition state accounts for the higher energy F + ICl scattering arising from the full range of impact parameters. The angular distribution for F + C2H5I shows peaking in the forward and backward directions which is consistent with a complex lifetime greater than one rotational period. The extended microcanonical theory accounts for scattering by a range of transition state configurations generated by internal rotation about the extended C-I bond for collisions at the largest impact parameters together with a phase space description of scattering at smaller impact parameters. The angular distribution for F + ICl at E ∼ 12 kJ mol-1 exhibits only mild peaking in the forward direction and a phase space description suffices for scattering from the full range of impact parameters. The strong dependence of the F + ICl reactive scattering on initial translational energy may be attributed to a contribution from migratory trajectories where the F atom interacts initially with the Cl atom of ICl. Nonmigratory trajectories involve the F atom interacting only with the I atom throughout, as in the F + C2H5I reaction.

Reactive scattering of a supersonic fluorine atom beam: F + C3H5I

Reactive scattering of F atoms with C3H5I molecules has been studied at an initial translational energy E ∼ 38 kJ mol-1 using a supersonic beam of F atoms seeded in He buffer gas and at E ∼ 12 kJ mol-1 with Ne buffer gas. The centre-of-mass angular distribution of IF scattering at E ∼ 38 kJ mol -1 shows peaking in the forward and backward directions which is consistent with reaction via a short-lived F-I-C3H5 complex with a lifetime less than one rotational period. The angular distribution at E ∼ 12 kJ mol-1 has a broad peak in the forward direction. The extended microcanonical theory shows that only a minor fraction of scattering at high initial translational energy may be identified with the precessional motion of the F-I-C3H5 product transition state formed in large impact parameter collisions. However, a major fraction of scattering at E ∼ 38 kJ mol-1 and all the scattering at E ∼ 12 kJ mol-1 may be described by phase space theory. The product translational energy distributions indicate that energy is...

Reactive scattering of a supersonic fluorine-atom beam: F + C2H5I, C3H7I, (CH3)2CHI

Reactive scattering of F atoms with C2H5I, C3H7I and (CH3)2CHI molecules has been studied at an initial translational energy E ≈ 38 kJ mol-1 using a supersonic beam of F atoms seeded in He buffer gas. Centre-of-mass angular distributions of IF scattering show peaking in the forward and backward directions, which is consistent with reaction via persistent F-I-C2H5, F-I-C3H7 and F-I-CH(CH3)2 complexes with lifetimes greater than two rotational periods. The extended microcanonical theory indicates that the angular distribution of reactive scattering for F + C2H5I arises from a range of transition state configurations generated by internal rotation about the extended C-I bond. The reactive scattering for F + C3H7I arises from a wide range of transition-state configurations, with sharply peaked forward and backward scattering from extended configurations and mildly peaked scattering from more contracted configurations. However, the range of transition-state configurations is much more limited for F + (CH3)2CHI. This distinction confirms the persistence of significant C-I bonding, which restricts the rotation of the IF and C2H5, C3H7 or (CH3)2CH fragments in these product transition states.

Preferred transition-state geometry in the reaction of fluorine atoms with iodomethane molecules

Reactive scattering of F atoms with CH3l molecules has been studied at an initial translational energy E≈ 34 kJ mol–1, using a supersonic beam of F atoms seeded in He buffer gas generated by a high-pressure microwave discharge source. The centre-of-mass angular distribution is represented by the Halpern–Strutinski function with a small constant term and a backward peak which is lower than the forward peak by a factor of ca. 0.75. The angular distribution of reactive scattering obtained from these measurements and the angular distribution previously determined by Farrar and Lee at lower initial translational energy E≈ 11 kJ mol–1 are both consistent with the microcanonical theory for dissociation of an F–l–CH3 complex via a bent transition state with a preferred interbond angle β≈ 140°.

The absorption line shape for a molecular system stochastically coupled to a phonon thermal reservoir

In this paper we extend earlier results regarding the relaxation properties of a molecular system, stochastically coupled to a phonon thermal reservoir, and interacting with a radiation field. The absorption line shape is derived in the white as well as the colored noise regime. Broadening, shift, and asymmetry are predicted. The expressions for the parameters, which characterize the features of the line, have an explicit dependence on the temperature, which our earlier, microcanonical theory did not include.

Some identities in isoperimetric problems in statistical mechanics and quantum mechanics

An identity is derived for the isoperimetric problem, which shows the close relation between the optimum value and the parameter involved in the constraint condition. The use of the identity is illustrated to finding the physical meaning of the Lagrange multipliers in the microcanonical theory of statistical mechanics. The identity is further generalized, and it is shown that the well-known Feynman theorem is a special case.

A statistical-theoretical investigation of the thermal rate coefficient and branching ratio for the reaction atomic oxygen + hydrogen cyanide .fwdarw. products

Using the BAC-MP4 potential surface parameters of Melius and Binkley, we have predicted the thermal rate coefficients for the two reactions: O + HCN ..-->.. NCO + H (a) and O + HCN ..-->.. NH + CO (b). Several levels of approximation are used in the theoretical treatment: a, canonical theory; b, canonical theory with Wigner tunneling correction; c, microcanonical theory (energy conserving); d, microcanonical/J-conservative theory (conserves both energy and angular momentum); e, microcanonical/J-conservative theory with one-dimensional tunneling. At high temperature the available experimental results are predicted accurately by even the crudest theoretical treatment (canonical theory). At lower temperature the theoretical predictions using the basic BAC-MP4 parameters are too low. However, adjustments to the BAC-MP4 energy barriers within their stated error limits lead to satisfactory agreement with experiment over the entire temperature range where experimental results are available (500 to 2500 K). The most important results of the investigation concern the dependence of the predictions on the level of approximation. Each successive refinement in the theory produces larger values of k/sub b/. The details of the theoretical treatment and comparisons with experiment are described in detail.

On the quantum statistical theory of relaxation in isolated spin systems II

The investigation of the irreversible behaviour of an isolated spin system, under taken in a previous paper, is continued. The question is studied in how far and in which way the transverse magnetic moments have to be taken into account for a correct description of the long time behaviour of the spin system. All the results obtained previously for the longitudinal magnetic moment are reproduced under the condition that the perturbation is spherically symmetrical in space. In order to determine the time corrections on the long time behaviour a formal solution is derived for the correlation tensor, valid for all times. It is shown that in the limit t → ∞ this expression yields a value, which is well in accordance with the predictions of microcanonical theory. In this connection the problem of the equivalence of the isolated and adiabatic susceptibilities is discussed. Furthermore, with the help of the formal solution for the correlation tensor, the line spectrum and line shapes are determined in parallel and perpendicular fields for several special cases.