Relaxation Time Scales Research Articles

Master equation lumping for multi-well potential energy surfaces: A bridge between ab initio based rate constant calculations and large kinetic mechanisms

Ab initio transition state theory-based master equation methodologies for the calculation of rate constants have gained enormous popularity in the past decades. Nevertheless, introducing these rate constants into large kinetic schemes is a non-trivial task when large potential energy surfaces (PESs) are investigated. To determine proper phenomenological rate constants it is in fact necessary to account for the formation of all the thermodynamically stable wells considered in the master equation (ME), even if most wells do not exhibit significant secondary reactivity. Moreover, reactions involving intermediates with lifetimes comparable to the rovibrational relaxation timescale can exhibit discontinuities both in the rate constants and in the number of thermodynamically stable wells across the investigated temperature and pressure ranges. In this work, we address these problems with a “master equation-based lumping” (MEL) approach specifically designed to process the output of ME calculations of multi-well PESs. Simple kinetic simulations allow identifying both intermediate wells with limited lifetime and isomers with similar reactivity. Then, equivalent rate constants for a smaller set of pseudospecies are derived so as to reproduce the kinetics of the detailed mechanism. Our methodology is independent of any experimental data or experience-based assumptions. The power of MEL is demonstrated with three case studies of increasing complexity, namely the PES for CH3COOH decomposition, and the portions of the C5H5OH and C10H10/C10H9 PESs accessed from C5H5 + OH and C5H5 + C5H5 recombination. This work constitutes the first systematic step addressing the robust integration of rate constants derived from ME simulations into global kinetic schemes and provides a useful approach for the entire chemical kinetics community filling the gap between detailed theoretical investigations of complex PESs and the development of detailed kinetic models.

What Markov State Models Can and Cannot Do: Correlation versus Path-Based Observables in Protein-Folding Models.

Markov state models (MSMs) have been widely applied to study the kinetics and pathways of protein conformational dynamics based on statistical analysis of molecular dynamics (MD) simulations. These MSMs coarse-grain both configuration space and time in ways that limit what kinds of observables they can reproduce with high fidelity over different spatial and temporal resolutions. Despite their popularity, there is still limited understanding of which biophysical observables can be computed from these MSMs in a robust and unbiased manner, and which suffer from the space-time coarse-graining intrinsic in the MSM model. Most theoretical arguments and practical validity tests for MSMs rely on long-time equilibrium kinetics, such as the slowest relaxation time scales and experimentally observable time-correlation functions. Here, we perform an extensive assessment of the ability of well-validated protein folding MSMs to accurately reproduce path-based observable such as mean first-passage times (MFPTs) and transition path mechanisms compared to a direct trajectory analysis. We also assess a recently proposed class of history-augmented MSMs (haMSMs) that exploit additional information not accounted for in standard MSMs. We conclude with some practical guidance on the use of MSMs to study various problems in conformational dynamics of biomolecules. In brief, MSMs can accurately reproduce correlation functions slower than the lag time, but path-based observables can only be reliably reproduced if the lifetimes of states exceed the lag time, which is a much stricter requirement. Even in the presence of short-lived states, we find that haMSMs reproduce path-based observables more reliably.

Numerical simulations of dark matter haloes produce polytropic central cores when reaching thermodynamic equilibrium

ABSTRACT Self-gravitating astronomical objects often show a central plateau in the density profile (core) whose physical origin is hotly debated. Cores are theoretically expected in N-body systems of maximum entropy, however, they are not present in the canonical N-body numerical simulations of cold dark matter (CDM). Our work shows that despite this apparent contradiction between theory and numerical simulations, they are fully consistent. Simply put, cores are characteristic of systems in thermodynamic equilibrium, but thermalizing collisions are purposely suppressed in CDM simulations. When collisions are allowed, N-body numerical simulations develop cored density profiles, in perfect agreement with the theoretical expectation. We compare theory and two types of numerical simulations: (1) when DM particles are self-interacting (SIDM) with enough cross-section, then the effective two-body relaxation time-scale becomes shorter than the Hubble time resulting in cored DM haloes. The haloes thus obtained, with masses from dwarf galaxies to galaxy clusters, collapse to a single shape after normalization, and this shape agrees with the polytropic density profile theoretically expected. (2) The inner radii in canonical N-body numerical simulations are always discarded because the use of finite-mass DM particles artificially increases the two-body collision rate. We show that the discarded radii develop cores which are larger than the employed numerical softening and have polytropic shape independently of halo mass. Our work suggests that the presence of cores in simulated (or observed) density profiles can used as evidence for systems in thermodynamic equilibrium.

Isomorphism between the electro-elastic modeling of the spin transition and Ising-like model with competing interactions: Elastic generation of self-organized spin states

Elastic modeling of spin-crossover materials has boomed remarkably these last years. Among these models, the electro-elastic model combining spin and lattice degrees of freedom showed good abilities of fair description of the thermodynamics and spin-crossover solids. In the present work, we explore a new treatment of this model based on a homogeneous description of the lattice spacing with well separate relaxation timescales for the lattice and spin state degrees of freedom. This description is analogous to the Born–Oppenheimer approximation and allows analytic treatment of the elastic part of the model, thus simplifying considerably the model resolution. As a result, we have been able to demonstrate the equivalence between the genuine electro-elastic model and an Ising-like Hamiltonian with competing long-range ferro-like and short-range (nearest neighbors and next-nearest neighbors along diagonals) antiferro-like interactions, whose relationship with the high-spin to low-spin misfit elastic energy has been established. This model generates intrinsic elastic frustration in the lattice, which leads to a rich variety of hysteretic first-order transitions made of one- two-, three-, or four-step behaviors. Complex self-organizations of the spin states are evidenced in the plateau regions in the form of checkerboard-like, stripes-like patterns, constituted of alternate high-spin and low-spin ferro-like stripes or alternate ferro high-spin (or low-spin) and antiferro-like chains, as well labyrinth structures.

Toward Using Seismic Interferometry to Quantify Landscape Mechanical Variations after Earthquakes

ABSTRACT In mountainous terrain, large earthquakes often cause widespread coseismic landsliding as well as hydrological and hydrogeological disturbances. A subsequent transient phase with high landslide rates has also been reported for several earthquakes. Separately, subsurface seismic velocities are frequently observed to drop coseismically and subsequently recover. Consistent with various laboratory work, we hypothesize that the seismic-velocity changes track coseismic damage and progressive recovery of landscape substrate, which modulate landslide hazard and hydrogeological processes, on timescales of months to years. To test this, we analyze the near-surface seismic-velocity variations, obtained with single-station high-frequency (0.5–4 Hz) passive image interferometry, in the epicentral zones of four shallow earthquakes, for which constraints on landslide susceptibility through time exist. In the case of the 1999 Chi-Chi earthquake, detailed landslide mapping allows us to accurately constrain an exponential recovery of landslide susceptibility with a relaxation timescale of about 1 yr, similar to the pattern of recovery of seismic velocities. The 2004 Niigata, 2008 Iwate, and 2015 Gorkha earthquakes have less-resolved constraints on landsliding, but, assuming an exponential recovery, we also find matching relaxation timescales, from ∼0.1 to ∼0.6 yr, for the landslide and seismic recoveries. These observations support our hypothesis and suggest that systematic monitoring of seismic velocities after large earthquakes may help constrain and manage the evolution of landslide hazard in epicentral areas. To achieve this goal, we end by discussing several ways to improve the link between seismic velocity and landscape mechanical properties, specifically by better constraining time-dependent near-surface strength and hydrogeological changes. Hillslopes displaying coseismic surface fissuring and displacement may be an important target for future geotechnical analysis and coupled to passive geophysical investigations.

Electronic noise of warm electrons in semiconductors from first principles

The ab-initio theory of low-field electronic transport properties such as carrier mobility in semiconductors is well-established. However, an equivalent treatment of electronic fluctuations about a non-equilibrium steady state, which are readily probed experimentally, remains less explored. Here, we report a first-principles theory of electronic noise for warm electrons in semiconductors. In contrast with typical numerical methods used for electronic noise, no adjustable parameters are required in the present formalism, with the electronic band structure and scattering rates calculated from first-principles. We demonstrate the utility of our approach by applying it to GaAs and show that spectral features in AC transport properties and noise originate from the disparate time scales of momentum and energy relaxation, despite the dominance of optical phonon scattering. Our formalism enables a parameter-free approach to probe the microscopic transport processes that give rise to electronic noise in semiconductors.

Activity pulses induce spontaneous flow reversals in viscoelastic environments.

Complex interactions between cellular systems and their surrounding extracellular matrices are emerging as important mechanical regulators of cell functions, such as proliferation, motility and cell death, and such cellular systems are often characterized by pulsating actomyosin activities. Here, using an active gel model, we numerically explore spontaneous flow generation by activity pulses in the presence of a viscoelastic medium. The results show that cross-talk between the activity-induced deformations of the viscoelastic surroundings and the time-dependent response of the active medium to these deformations can lead to the reversal of spontaneously generated active flows. We explain the mechanism behind this phenomenon based on the interaction between the active flow and the viscoelastic medium. We show the importance of relaxation time scales of both the polymers and the active particles and provide a phase space over which such spontaneous flow reversals can be observed. Our results suggest new experiments investigating the role of controlled pulses of activity in living systems ensnared in complex mircoenvironments.

The effect of fluid viscoelasticity in lubricated contacts in the presence of cavitation

In this work we study the influence of fluid viscoelasticity on the performance of lubricated contacts in the presence of cavitation. Several studies of viscoelastic lubricants have been carried out, but none of them have considered the possibility of the presence of cavitation. To describe the effect of viscoelasticity, we use the Oldroyd-B model. By assuming that the product between ϵ, i.e. the ratio between vertical and horizontal length scales, and the Weissenberg number (Wi), i.e. the ratio between polymer relaxation time and flow time scale, is small, we can linearise the viscoelastic thin film equations, following the approach pioneered by "Tichy, J., 1996, Non-Newtonian lubrication with the convected Maxwell model." Consequently, the zeroth-order in ϵWi corresponds to a Reynolds equation modified to describe also the film cavitation through the mass-conserving Elrod-Adams model. We consider the flow of viscoelastic lubricants using: (i) a cosine profile representing a journal bearing unwrapped geometry, and (ii) a pocketed profile to model a textured surface in lubricated contacts. The introduction of viscoelasticity decreases the length of cavitated region in the cosine profile due to the increasing pressure distribution within the film. Consequently, the load carrying capacity increases with Wi by up to 50% in the most favorable condition, confirming the beneficial influence of the polymers in bearings. On the other hand for the pocketed profile, results show that the load can increase or decrease at higher Wi depending on the texture position in the contact. The squeeze flow problem between two plates is also modeled for viscoelastic lubricants considering an oscillating top surface. For this configuration a load reduction is observed with increasing Wi due to the additional time needed to reform the film at high Wi. Furthermore, if viscoelastic effects increase, the cavitation region widens until reaching a value of Wi for which a full-film reformation does not occur after the initial film rupture.

Molecular dynamics simulations vs field-cycling NMR relaxometry: Structural relaxation mechanisms in the glass-former glycerol revisited.

We combine field-cycling (FC) relaxometry and molecular dynamics (MD) simulations to study the rotational and translational dynamics associated with the glassy slowdown of glycerol. The 1H NMR spin-lattice relaxation rates R1(ω) probed in the FC measurements for different isotope-labelled compounds are computed from the MD trajectories for broad frequency and temperature ranges. We find high correspondence between experiment and simulation. Concerning the rotational motion, we observe that the aliphatic and hydroxyl groups show similar correlation times but different stretching parameters, while the overall reorientation associated with the structural relaxation remains largely isotropic. Additional analysis of the simulation results reveals that transitions between different molecular configurations are slow on the time scale of the structural relaxation at least at sufficiently high temperatures, indicating that glycerol rotates at a rigid entity, but the reorientation is slower for elongated than for compact conformers. The translational contribution to R1(ω) is well described by the force-free hard sphere model. At sufficiently low frequencies, universal square-root laws provide access to the molecular diffusion coefficients. In both experiment and simulation, the time scales of the rotational and translational motions show an unusually large separation, which is at variance with the Stokes-Einstein-Debye relation. To further explore this effect, we investigate the structure and dynamics on various length scales in the simulations. We observe that a prepeak in the static structure factor S(q), which is related to a local segregation of aliphatic and hydroxyl groups, is accompanied by a peak in the correlation times τ(q) from coherent scattering functions.

Cloud droplet diffusional growth in homogeneous isotropic turbulence: bin microphysics versus Lagrangian super-droplet simulations

Abstract. The increase in the spectral width of an initially monodisperse population of cloud droplets in homogeneous isotropic turbulence is investigated by applying a finite-difference fluid flow model combined with either Eulerian bin microphysics or a Lagrangian particle-based scheme. The turbulence is forced applying a variant of the so-called linear forcing method that maintains the mean turbulent kinetic energy (TKE) and the TKE partitioning between velocity components. The latter is important for maintaining the quasi-steady forcing of the supersaturation fluctuations that drive the increase in the spectral width. We apply a large computational domain (643 m3), one of the domains considered in Thomas et al. (2020). The simulations apply 1 m grid length and are in the spirit of the implicit large eddy simulation (ILES), that is, with small-scale dissipation provided by the model numerics. This is in contrast to the scaled-up direct numerical simulation (DNS) applied in Thomas et al. (2020). Two TKE intensities and three different droplet concentrations are considered. Analytic solutions derived in Sardina et al. (2015), valid for the case when the turbulence integral timescale is much larger than the droplet phase relaxation timescale, are used to guide the comparison between the two microphysics simulation techniques. The Lagrangian approach reproduces the scalings relatively well. Representing the spectral width increase in time is more challenging for the bin microphysics because appropriately high resolution in the bin space is needed. The bin width of 0.5 µm is only sufficient for the lowest droplet concentration (26 cm−3). For the highest droplet concentration (650 cm−3), an order of magnitude smaller bin size is barely sufficient. The scalings are not expected to be valid for the lowest droplet concentration and the high-TKE case, and the two microphysics schemes represent similar departures. Finally, because the fluid flow is the same for all simulations featuring either low or high TKE, one can compare point-by-point simulation results. Such a comparison shows very close temperature and water vapor point-by-point values across the computational domain and larger differences between simulated mean droplet radii and spectral width. The latter are explained by fundamental differences in the two simulation methodologies, numerical diffusion in the Eulerian bin approach and a relatively small number of Lagrangian particles that are used in the particle-based microphysics.

Rapid Charge Separation Boosts Solar Hydrogen Generation at the Graphene-MoS2 Junction: Time-Domain Ab Initio Analysis.

Transition metal dichalcogenides and graphene hybrids hold great promise for photovoltaics and photocatalysts. Using a combination of time-domain density functional theory and nonadiabatic molecular dynamics, we investigate the interplay between forward and backward electron transfer (ET), as well as energy relaxation in a van der Waals graphene-MoS2 heterojunction. We demonstrated that built-in potential formed at the polarized interface produces charge separation upon photoexcitation. The electron left on graphene is injected into MoS2 on an ultrafast time scale, which is notably faster than energy losses to heat regardless of the initial state energy. Once the electron is relaxed to the conduction band edge state of MoS2, it transfers back and recombines with the hole remaining on graphene on ultrafast time scales by considering quantum transitions among multiple k points. The obtained time scales for ET, back-ET, and energy relaxation agree well with experimental data. The study reveals that ET that is faster than energy loss makes the graphene-MoS2 heterojunction efficient for optoelectronic applications.

Photoinduced Dynamics of (CH3NH3)4Cu2Br6 Thin Films Indicating Efficient Triplet Photoluminescence.

Hybrid organic-inorganic halogenidocuprates based on copper(I) represent materials with rich structural diversity and high photoluminescence (PL) quantum yield, yet the mechanism responsible for their efficient, strongly Stokes-shifted emission is still unclear. Here we report the successful preparation of (CH3NH3)4Cu2Br6 thin films with a zero-dimensional molecular salt structure featuring "isolated" [Cu2Br6]4- ions. Time-resolved broadband PL measurements provide an excited-state lifetime of 114 μs at 298 K. Results from femto- to microsecond UV-vis-NIR transient absorption experiments combined with DFT/TDDFT calculations suggest the formation of a long-lived structurally relaxed triplet species through intersystem crossing (61 ps), which almost exclusively decays by phosphorescence. In addition, time scales for structural relaxation and cooling processes are extracted from a global kinetic analysis of the transient spectra. Calculations for the isolated [Cu2Br6]4- anion and the (CH3NH3)4Cu2Br6 crystal suggest a strong impact of the crystal environment on the structure of the anion.

Energy relaxation path of excited free OH vibration at an air/water interface revealed by nonequilibrium ab initio molecular dynamics simulation.

Nonequilibrium ab initio molecular dynamics (NE-AIMD) simulations are conducted at an air/water interface to elucidate the vibrational energy relaxation path of excited non-hydrogen-bonded (free) OH. A recent time-resolved vibrational sum frequency generation (TR-VSFG) spectroscopy experiment revealed that the relaxation time scales of free OH at the surface of pure water and isotopically diluted water are very similar to each other. In the present study, the dynamics of free OH excited at the surface of pure water and deuterated water are examined with an NE-AIMD simulation, which reproduces the experimentally observed features. The relaxation paths are examined by introducing constraints for the bonds and angles of water molecules relevant to specific vibrational modes in NE-AIMD simulations. In the case of free OH relaxation at the pure water surface, stretching vibrational coupling with the conjugate bond makes a significant contribution to the relaxation path. In the case of the isotopically diluted water surface, the bend (HOD)-stretching (OD) combination band couples with the free OH vibration, generating a relaxation rate similar to that in the pure water case. It is also found that the reorientation of the free OH bond contributes substantially to the relaxation of the free OH vibrational frequency component measured by TR-VSFG spectroscopy.

Role of integrated noise in pathway-specific signal propagation in feed-forward loops

Cells impose optimal noise control mechanism in diverse situations to cope with distinct environmental cues. Sometimes, it is desirable for the cell to utilize fluctuations for noise-driven processes. In other cases, noise can be harmful to the cell to show optimal fitness. It is, therefore, important to unravel the noise propagation mechanism inside the cell. Such noise controlling mechanism is accomplished by using gene transcription regulatory networks. One such gene regulatory network is feed-forward loop, having three regulatory nodes S, X and Y. Here, we consider the most abundant type 1 of coherent and incoherent feed-forward loops with both OR and AND logic functions, forming four different architectures. In OR logic function, the functions representing S and X act additively for the regulation of Y, while in AND logic function, the same functions (S and X) act multiplicatively for the regulation of Y. Measurement of susceptibility of the signal at output Y is done using elasticity of each regulation in FFLs. Using susceptibility, we demonstrate the nature of pathway integration by which one-step and two-step pathways get overlapped. The integration type is competitive for motifs having OR gate, while it is noncompetitive for the same with AND gate. The pathway integration property explains the output noise behavior of the motifs properly but cannot infer about the mechanism by which the upstream noise propagates to output. To account this, the total output noise is decomposed, which results in integrated noise as an additional noise source along with pathway-specific noise components. The integrated noise is found to appear as a consequence of integration between the pathways and has different functional characteristics explaining noise amplification and noise attenuation property of coherent and incoherent feed-forward loops, respectively. The noise decomposition also quantifies the contribution of different noise sources toward total noise. Finally, the noise propagation is being tuned as a function of input signal noise and its time scale of fluctuations, which shows considerable intrinsic noise strength and relatively slow relaxation time scale causes a higher degree of noise propagation in FFLs.

Kinetic and Calorimetric Fragility of Chalcogenide Glass-Forming Liquids: Role of Shear vs Enthalpy Relaxation.

The kinetic and calorimetric fragility indices m of binary As-Se and Se-Te chalcogenide liquids with a wide range of fragility are determined using a combination of parallel plate rheometry, beam bending viscometry, and conventional differential scanning calorimetry (DSC). It is shown that both sets of measurements lead to consistent m values only if the validity of the assumptions often implicit in the methodology for the estimation of m are considered. These assumptions are (i) the glass transition temperature Tg corresponds to a viscosity of ∼1012 Pa s and (ii) enthalpy and shear relaxation time scales τen and τshear are comparable near Tg. Both assumptions are shown to be untenable for highly fragile liquids, for which modulated DSC studies demonstrate that τen ≫ τshear near Tg. In these cases, the above-mentioned assumptions are shown to lead to consistently higher values for the kinetic fragility compared to its calorimetric counterpart.

Chiral switching and dynamic barrier reductions in artificial square ice

Collective dynamics in lithographically-defined artificial spin ices offer profound insights into emergent correlations and phase transitions of geometrically-frustrated Ising spin systems. Their temporal and spatial evolution are often simulated using kinetic Monte Carlo (kMC) simulations, which rely on the precise knowledge of the switching barriers to obtain predictive results in agreement with experimental observations. In many cases, however, the barriers are derived from simplified assumptions only, and do not take into account the full physical picture of nanomagnetic switching. Here we describe how the immediate magnetic square- or kagome-ice environment of a nanomagnet reversing via quasi-coherent rotation can induce clockwise and counter-clockwise switching channels with different barrier energies. This energy splitting for chiral reversal channels can be sizeable and, as string-method micromagnetic simulations show, is relevant for artificial spin ice systems made of both exchange- as well as magnetostatically-dominated units. Due to the barrier splitting and further reductions due to non-uniform reversal, transition rates can be exponentially enhanced by several orders of magnitude compared to mean-field predictions, especially in the limit of rare switching events where thermal excitation is less likely. This leads to significantly faster relaxation time scales and modified spatial correlations. Our findings are thus of integral importance to achieve realistic kMC simulations of emergent correlations in artificial spin systems, magnonic crystals, or the evolution of nanomagnetic logic circuits.

Johari–Goldstein Heterogeneous Dynamics in a Model Polymer

The heterogeneous character of the Johari-Goldstein (JG) relaxation is evidenced by molecular-dynamics simulation of a model polymer system. A double-peaked evolution of dynamic heterogeneity (DH), with maxima located at JG and structural relaxation time scales, is observed and mechanistically explained. {The short-time single-particle displacement during JG relaxation weakly correlates with the long-time one observed during structural relaxation}.

Parameter estimation of a two-component neutron star model with spin wandering

ABSTRACT It is an open challenge to estimate systematically the physical parameters of neutron star interiors from pulsar timing data while separating spin wandering intrinsic to the pulsar (achromatic timing noise) from measurement noise and chromatic timing noise (due to propagation effects). In this paper, we formulate the classic two-component, crust-superfluid model of neutron star interiors as a noise-driven, linear dynamical system and use a state-space-based expectation–maximization method to estimate the system parameters using gravitational-wave and electromagnetic timing data. Monte Carlo simulations show that we can accurately estimate all six parameters of the two-component model provided that electromagnetic measurements of the crust angular velocity and gravitational-wave measurements of the core angular velocity are both available. When only electromagnetic data are available, we can recover the overall relaxation time-scale, the ensemble-averaged spin-down rate, and the strength of the white-noise torque on the crust. However, the estimates of the secular torques on the two components and white-noise torque on the superfluid are biased significantly.

Periodic quenches across the Berezinskii-Kosterlitz-Thouless phase transition

The quenched dynamics of an ultracold homogeneous atomic two-dimensional Bose gas subjected to periodic quenches across the Berezinskii-Kosterlitz-Thouless (BKT) phase transition are discussed. Specifically, we address the effect of periodic cycling of the effective atomic interaction strength between a thermal disordered state above, and a highly ordered state below the critical BKT interaction strength, by means of numerical simulations of the stochastic projected Gross-Pitaevskii equation. Probing the emerging dynamics as a function of the frequency of sinusoidal driving from low to high frequencies reveals diverse dynamical features, including phase-lagged quasi adiabatic reversible condensate formation, resonant excitation consistent with an intrinsic system relaxation timescale, and gradual establishment of dynamically-recurring or time-averaged non-equilibrium states with enhanced coherence which are neither condensed, nor thermal. Our study paves the way for experimental observation of such driven non-equilibrium ultracold superfluid states.

Solvation dynamics in electronically polarizable solvents: Theoretical treatment using solvent-polarizable three-dimensional reference interaction-site model theory combined with time-dependent density functional theory.

The theory of solvation structure in an electronically polarizable solvent recently proposed by us, referred to as the "solvent-polarizable three-dimensional reference interaction-site model theory," is extended to dynamics in this study through the combination with time-dependent density functional theory. Test calculations are performed on model charge-transfer systems in water, and the effects of electronic polarizability on solvation dynamics are examined. The electronic polarizability slightly retards the solvation dynamics. This is ascribed to the decrease in the curvature of the nonequilibrium free energy profile along the solvation coordinate. The solvent relaxation is bimodal, and the faster and the slower modes are assigned to the reorientational and the translational modes, respectively, as was already reported by the surrogate theory combined with the site-site Smoluchowski-Vlasov equation. The relaxation path along the solvation coordinate is a little higher than the minimum free energy path because the translational mode is fixed in the time scale of the reorientational relaxation.