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More specifically, our aim is to examine the nature of discrete solitons for the AL-type lattice nonlinearity, [MATH] , with different values of [MATH] (characterizing also the structure of these discrete solitons by means of a variational approach) and to explore whether these modes change their stability with the var... |
[MATH] (or [MATH] , respectively) norm vs. the soliton’s intrinsic frequency dependence induces a change of its stability book .Finally, we explore the full stability (via the spectral stability analysis) and evolution of instabilities in the model, by means of numerical computations. This allows us, among other featur... |
Our presentation is structured as follows. In section II, we offer our analytical considerations, including the model setup, the present variant of the VK criterion, and the variational approximation. In section III, we show numerical computations concerning both the existence and spectral stability of discrete soliton... |
II Analytical Considerations II.1 The general Setup, conservation laws, and stability analysis Our generalized AL model with the power-law nonlinearity is introduced as |
[EQUATION] Here, [MATH] is the (linear) coupling constant between adjacent sites, while [MATH] represents, as defined above, the nonlinearity power. In what follows below, we fix [MATH] by means of obvious rescaling, considering the focusing sign of the nonlinearity ( [MATH] ). This model can be thought of as a discret... |
in the 1D case considered here) of the collapse phenomenology in discrete vs. continuum models. With these motivations in mind, we will construct localized states of the model, investigate their stability through computation of eigenvalues of small perturbations, and numerically examine evolution of unstable states. |
An important property of Eq. ( ) is that it preserves a suitably defined norm-like quantity. The derivation of this property, that we will develop here, is generally applicable to nonlinear lattice equations of the form |
[EQUATION] where [MATH] may be an arbitrary real function of [MATH] . We assuming that [MATH] (which should revert to the usual [MATH] norm in the continuum limit) depends only on the on-site intensity [MATH] |
[EQUATION] From here it follows that [EQUATION] where Eq. ( ) is used to substitute [MATH] , and [MATH] stands for the derivative of [MATH] . Thus, the selection of [MATH] leads [MATH] , due to the telescopic nature of the resulting summation. For the particular case of [MATH] , this results in |
[EQUATION] where [MATH] is the density associated with the conserved quantity at each node, and [MATH] is the Gauss hypergeometric function. In the special case of [MATH] , it is straightforward to see that this falls back to the simple expression relevant for the cubic integrable AL model cai [MATH] (up to a [MATH] -d... |
A further important observation for Eq. ( ) is that a natural way to analyze the system can be to explore its similarity to the integrable case of |
[MATH] and the perturbation theory around it cai review . In particular, we conclude that the Hamiltonian of the general AL equation remains the same as in the integrable case, |
[EQUATION] being a conserved quantity too. It is also relevant to define the Poisson brackets corresponding to this Hamiltonian: |
[EQUATION] where [MATH] and [MATH] are two arbitrary functionals. This definition leads to [MATH] , while [MATH] . The respective Hamiltonian form of Eq. ( ) is |
[EQUATION] In our numerical computations, we sought for stationary solutions with frequency [MATH] , in the form of [MATH] with real [MATH] obeying the following set of algebraic equations: |
[EQUATION] The linear stability was explored by considering perturbations with infinitesimal amplitude [MATH] around the stationary solutions. To this end, we substitute ansatz [MATH] in Eq. ( ), and then solve the ensuing linearized eigenvalue problem: [MATH] , with matrix |
[EQUATION] where submatrices [MATH] and [MATH] in Eq. ( 10 ) are composed of the following entries: [EQUATION] [EQUATION] The stationary solutions are spectrally stable if the eigenvalue problem produces purely imaginary [MATH] . On the contrary, the existence of eigenvalues with nonzero real parts implies instability.... |
II.2 The stationary variational approximation Before turning to numerical computations, it is natural to apply a variational approximation (VA) progopt to the stationary equation ( ). This will allow us to examine how well numerical solutions, that are produced in the next section, can be approximated by the simple (ex... |
[EQUATION] and thus rewrite Eq. ( ) as [EQUATION] which may generate bright solitons for [MATH] The Lagrangian from which Eq. ( 14 ) can be derived is |
[EQUATION] The integral in this expression can be formally expressed in terms of the hypergeometric function, similar to Eq. ( ), but such a formula is not really needed. |
The simplest ansatz for the stationary soliton follows the pattern of Ref. malomed [EQUATION] with [MATH] . Then, the first term in Eq. ( 15 ) is easily calculated, following the substitution of ansatz ( 16 ): |
[EQUATION] The variational (Euler-Lagrange) equations intended to provide stationary discrete solitons solutions, derived from Lagrangian ( 15 ), in which ansatz ( 16 ) is substituted, read: |
[EQUATION] The explicit form of the resulting equations is: [EQUATION] The system of Eqs. ( 19 ) and ( 20 ) can be solved numerically for variable [MATH] |
and [MATH] if [MATH] and [MATH] are given. Although the solution of this system is still performed numerically, hence our approach is not fully analytical, the VA provides a significant insight into the problem, as it converts the infinite-dimensional original system into a much simpler system of two equations ( 19 ) a... |
III Numerical Results First of all, we consider the dependence of the stability of 1-site (site-centered) and 2-site (inter-site centered) discrete solitons on the nonlinearity power [MATH] for fixed frequency [MATH] and coupling [MATH] . Such solutions, e.g., for both kinds of modes at [MATH] [MATH] and [MATH] are dep... |
As it is well known book , in the case of the cubic ( [MATH] ) AL equation, there are two eigenvalue pairs at zero, as a consequence of the translational and phase (gauge) invariance of the model. One of these invariances, namely the effective translational invariance, is broken in the non-integrable case, [MATH] . We ... |
[MATH] . The scenario is quite different for [MATH] , where the situation is reversed for 1-site and 2-site modes. In particular, in that case, the translational mode moves along the real (imaginary) axis for the 1-site (2-site) soliton, giving rise to exchange of the stability between them. It is worthy to note that a... |
(recall that [MATH] corresponds to the LHY nonlinearity in the two-component BEC Grisha1 ). All these phenomena are summarized in Fig. |
Secondly, we have analyzed the effect of varying the coupling constant on the critical values [MATH] and [MATH] . The approach helps to reach the continuum limit, [MATH] . Figure |
shows the dependence of [MATH] on [MATH] for [MATH] . One can observe that the minimum value of [MATH] is [MATH] , a value that is attained at [MATH] ; furthermore, [MATH] tends to [MATH] |
when [MATH] and [MATH] . The latter retrieves the well-known continuum limit of the corresponding eigenvalue bifurcation towards the collapse sulem . As mentioned in the above paragraph, the critical value [MATH] is independent of [MATH] and is equal to [MATH] Interestingly, the spectrum of solitons for [MATH] is the s... |
[MATH] except for the occurrence of two (pairs of) localized eigenmodes, one at the top and another one at the bottom of the linear modes band. Notice that both 1-site and 2-site solitons possess the same spectrum,as illustrated in Fig. , and an associated neutral mode, as confirmed in Fig. . Notice, however, that the ... |
Next we test the validity of the VK criterion in the present setting (see, e.g., book for its application to DNLS equations). It suggests that the change of monotonicity of the frequency dependence of the norm-like conserved quantity, defined as per Eq. ( ) amounts to a change of stability. To examine this, we take 1-s... |
and [MATH] . The result is shown in Fig. , where one can indeed see the direct correlation between the slope of [MATH] dependence and the actual stability, as expected from the VK criterion, and from the general Grillakis-Shatah-Straus theory grillakis ; see Ref. kapprom for a recent overview. More specifically, the di... |
We have also compared the predictions of the VA, presented in subsection II.2 , with the results stemming from numerically exact (up to a prescribed accuracy) calculations for the discrete solitons. Figure shows the amplitude, [MATH] , and the conserved quantity [MATH] of 1-site solitons versus [MATH] , for three selec... |
More specifically, in the case of [MATH] the power is predicted to be a monotonic function of [MATH] by the VA, while the monotonicity is not held by the numerical solution. This inaccuracy is a consequence of the oversimplified ansatz adopted in Eq. ( 16 ); on the other hand, a more sophisticated ansatz would not be a... |
Finally, we mention the main features of unstable discrete solitons evolution for different typical examples of the instability that are identified above. Figures and show the outcome of the evolution for unstable 1-site solitons with [MATH] : it is observed that the soliton density at [MATH] [MATH] , deviates from the... |
increases with [MATH] , and for [MATH] it leads to computational overflow. Beyond this limit, despite using multiple algorithms, we have been unable to converge to a solution. Because of this, we cannot go beyond this value of [MATH] in our simulations. Our phenomenology basically involves an extremely fast increasing ... |
The dynamics of unstable 2-site solitons with [MATH] is somewhat different when [MATH] is close to 1. If [MATH] is close enough to 1 (as e.g. [MATH] for [MATH] ), the soliton becomes mobile but, if [MATH] is high enough (as e.g. [MATH] for [MATH] ), the soliton gets pinned and its profile oscillates between a 1-site an... |
too, decreasing with the increase of [MATH] (see Fig. 10 ). The high density fraction that concentrates at [MATH] experiences a similar trend to the observed in the 1-site case. |
It is also worthy to mention that unstable 1-site solitons with [MATH] feature mobility in the underlying lattice (not shown here in detail), which is a consequence of the stability-exchange bifurcation associated with the translational perturbation mode. |
IV Conclusions and future challenges In the present work we have explored a non-integrable generalization of the AL (Ablowitz-Ladik) model, bearing the power-law nonlinearity. The model by itself is interesting for a variety of reasons, such as the discretization of the continuum problem with the power-law nonlinearity... |
Our analysis has revealed essential features of the model. We have identified its conservation laws, including an unprecedented form of the conserved quantity (which falls back to the well-known logarithmic expression for the AL norm at [MATH] ) and the Hamiltonian. We have demonstrated how the modified norm-like quant... |
The present study paves the way for numerous questions worthwhile of further examination. Elaborating a better variational ansatz, if it may be (semi-) analytically tractable, will certainly help to develop more accurate understanding of the model’s features. An exploration of stability switchings, such as the one at t... |
Acknowledgements. J.C.-M. thanks the financial support from MAT2016-79866-R project (AEI/FEDER, UE). P.G.K. acknowledges the support by NPRP grant # [9-329-1-067] from Qatar National Research Fund (a member of Qatar Foundation). The work of B.A.M. was supported, in part, by the Israel Science Foundation, through grant ... |
# Source: arxiv 1806.11070 # Title: The effect of memory and active forces on transition path times distributions # Sections: all # Downloaded: 2026-03-03T05:16:00.972139+00:00 |
\alsoaffiliation Beijing Computational Science Research Center, No.10 East Xibeiwang Road, Beijing 100193, China \alsoaffiliation PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan |
\alsoaffiliation KU Leuven, Institute for Theoretical Physics, Celestijnenlaan 200D, 3001 Leuven, Belgium The effect of memory and active forces on transition path times distributions |
Abstract An analytical expression is derived for the transition path time distribution for a one-dimensional particle crossing of a parabolic barrier. Two cases are analyzed: (i) A non-Markovian process described by a generalized Langevin equation with a power-law memory kernel and (ii) a Markovian process with a noise... |
Introduction Biomolecular folding involves structural transitions of various time- and lengthscales. A simplified description of this process employs a single reaction coordinate performing stochastic dynamics along a free energy landscape. In the case of a two state folding, the folded and unfolded states correspond t... |
Anomalous dynamics is ubiquitous in macromolecular systems as polymers, as it is known from many examples 21 22 23 24 25 26 27 This dynamics is characterized by a mean-square displacement of a suitable reaction coordinate scaling as [MATH] , with [MATH] . The analysis of the effect of an underling anomalous dynamics on... |
Generalities We consider a particle performing a stochastic dynamics on an inverted parabolic potential barrier [MATH] , with [MATH] . At time |
[MATH] the particle starts from a point [MATH] . Transition paths are those paths reaching [MATH] at the right side of the barrier without recrossing [MATH] and [MATH] . To compute the distribution of their duration one should solve the Langevin equation imposing absorbing boundary conditions in [MATH] and [MATH] . Fre... |
[MATH] the inverse temperature. This is because the probability of multiple crossings in [MATH] is negligible for high barriers. |
In Ref. 18 the TPT distribution was calculated for a Markovian particle with inertia. It was shown that both in the inertial and overdamped cases the TPT distribution assumes the general form 18 |
[EQUATION] where [MATH] and [EQUATION] In the previous equation [EQUATION] are the mean and variance of the process. In the overdamped case the function [MATH] assumes a simple form |
[EQUATION] where [MATH] and [MATH] is the friction coefficient. The [MATH] in the inertial case is more complex and is given in Ref. 18 |
For short times [MATH] , which diverges as a consequence of the the initial condition [MATH] , implying [MATH] . This leads to a TPT distribution vanishing with an essential singularity as [MATH] . In the Markovian case the behavior was found to be different in the overdamped [MATH] and inertial [MATH] cases 18 (here [... |
[MATH] the diffusion coefficient). At long times [MATH] converges to a constant in both cases, while its derivative decays exponentially [MATH] , where [MATH] |
is the longest relaxation time of the process ( [MATH] in the overdamped limit ( )). This leads to an exponential decay [MATH] for the long time behavior of the distribution both in the overdamped and inertial case 18 . In the next Section we compute [MATH] for a process with correlated and active noise and discuss the... |
Memory effects in transition path times A reaction coordinate [MATH] is by definition a slow variable for which standard statistical mechanical arguments show that its time evolution is given in terms of a generalized Langevin equation 29 . For a parabolic barrier in the overdamped case this equation takes the form |
[EQUATION] Here [MATH] is a memory kernel. The noise [MATH] is assumed to be a Gaussian process with average zero and a correlation that in equilibrium is related to [MATH] by the fluctuation-dissipation theorem |
[EQUATION] We focus here on a power law memory kernel [EQUATION] where [MATH] and where, following Ref. 27 we define the generalized friction coefficient as [MATH] . In the limit [MATH] , the [MATH] function in the denominator becomes singular and [MATH] , i.e. one recovers the Markovian (memoryless) dynamics. Power la... |
[MATH] the effects of memory on the motion of a reaction coordinate can be neglected, these are strongly influencing the polymer dynamics for [MATH] . Polymers have a memory kernel [MATH] that, for [MATH] can be well approximated by a power law 30 31 . This power law behavior is a characteristic of systems with a broad... |
The generalized Langevin equation ( ) with the kernel ) is a linear equation which can be solved using Laplace transforms. The initial condition is [MATH] . As explained in the previous section we do not impose specific boundary conditions in [MATH] , an approximation which is good for steep barriers |
[MATH] . The solution of ( ) is (for details see Appendix [EQUATION] where we introduced the functions [EQUATION] [MATH] is the characteristic rate of the process) and where |
[EQUATION] is the Mittag-Leffler function 32 We assume that the noise [MATH] is Gaussian, and since the Langevin equation ( ) is linear, we conclude that also [MATH] is Gaussian. Hence Eqs. ( ) and ( ) apply. One has for the average |
[EQUATION] while the variance is given by [EQUATION] (details of the calculations are in Appendix ). Plugging in ( 13 ) and ( 14 ) in ( ) we get: |
[EQUATION] This result generalizes the memoryless case ( ), which is recovered in the limit [MATH] since [MATH] Figure shows plots of the transition path distribution |
[MATH] for three different values of [MATH] and for two different values of [MATH] . The transition path time (in dimensionless units) decreases with decreasing [MATH] . We now look at the behavior of [MATH] for small and large times which can be obtained from the corresponding behavior of the Mittag-Leffler functions.... |
[EQUATION] which implies that [MATH] vanishes exponentially, as was the case for the Markovian model 18 . This is in contrast with the conclusions of a recent paper 20 where a stretched exponential decay was found. However, the results of that paper where obtained from a Fokker-planck equation for systems with memory t... |
For [MATH] , one has that [MATH] (see ( 51 )) from which it follows that [MATH] . The behavior of the transition path time distribution for early times is therefore determined by the essential singularity in [MATH] . Hence |
[EQUATION] We see that the early time behavior does depend on [MATH] and that the exponent could be determined from a straight line fit to a log-log plot of [MATH] versus [MATH] . In Fig. |
we have made such a plot for [MATH] and [MATH] . We find the expected power law behavior for [MATH] . An integration of the TPT distribution shows however that the probability that the transition path time is in this regime is extremely low ( [MATH] ). We therefore conclude that it is experimentally impossible to deter... |
[MATH] from the early time behavior of the TPT. We look next at the behavior of the average transition path time following the same procedure as outlined in Ref. 18 . As [MATH] is a monotonic decreasing function of [MATH] for the calculation it is convenient to perform a change of variable: |
[EQUATION] The integral in the numerator can not be performed exactly. We can however get an approximation for [MATH] sufficiently large. In that limit the integrals in ( 18 ) are determined by the large [MATH] -limit of [MATH] The average TPT is then given by (for details, see Appendix |
[EQUATION] where [MATH] is the Euler-Mascheroni constant (in the Markovian limit [MATH] this expression coincides with that previously obtained by Szabo 34 ). In Fig. , we have plotted the result of a numerical calculation of the average transition path time as a function of [MATH] using the full expression for [MATH] ... |
[MATH] , decreases with decreasing [MATH] as was already evident from the plots in Fig. . The most likely transition path time [MATH] , corresponding to the maximum of a distribution is (see Appendix) |
[EQUATION] and show a similar logarithmic dependence on the barrier height as the average ( 19 ). The comparison between the analytical expression 20 ) and the numerical calculation of the maximum is shown in Fig. as dashed lines. |
Transition path times in the presence of active forces The folding of a biopolymer in vivo takes place in an environment which is out of equilibrium due to the action of various ATP-dependent active processes within a cell. These processes are known to modify the dynamics of various ”probes” like microspheres 35 36 37 ... |
In a recent study on the behavior of active Brownian particles near soft walls, the dynamics of a semiflexible polymer immersed in an environment of such particles was investigated 49 . Active Brownian particles move in a direction [MATH] which is subject to rotational Brownian diffusion. The force they produce on a (f... |
Inspired by these two examples of folding in a non-equilibrium environment, it may be of interest to study also the effect of active forces on transition path times. We start from the Langevin equation |
[EQUATION] Here [MATH] is now a Markovian thermal force while [MATH] is the active noise which assume to have an exponential correlation. |
[EQUATION] We also take [MATH] . The coefficient [MATH] measures the strength of the active force. There is no associated friction force so that ( 21 ) describes a system out of equilibrium. |
The solution of ( 21 ) with initial condition [MATH] is [EQUATION] From this we find that the deterministic motion is [EQUATION] |
while the variance of the position is given by [EQUATION] It is convenient to describe the escape over the parabolic in terms of an effective temperature as was done in a study of the motion of colloids in active bath of bacteria and in the presence of a confining harmonic potential 50 |
Asymptotically [MATH] goes to a constant which can be used to define this effective temperature [MATH] [EQUATION] The effective temperature takes over the role of the physical temperature in the transition path time distribution. Going through the calculations of Ref. 18 we find that in this case |
[EQUATION] where [MATH] is given by ( ) and [MATH] From these results one can find that at early times, the transition path time is again governed by the essential singularity in [MATH] whose form is not modified by the active forces. The late time decay is governed by [MATH] which decays exponentially. The only change... |
[EQUATION] Finally, in the expression for the average transition path time, the effective temperature also replaces the real temperature |
[EQUATION] Since [MATH] , the addition of active forces leads to a decrease of the average transition path time. In Fig. we have plotted some distributions where it can be seen that indeed the average transition path time decreases if the effective temperature (here tuned by changing [MATH] at fixed [MATH] ) increases. |
As can be seen from the expression of the effective temperature, the dependence on [MATH] is weak once it becomes bigger then [MATH] (the other time scale in problem). This can also be seen in Fig. where [MATH] is fixed and [MATH] is increased from values below to values above [MATH] |
Discussion Conformational transitions of molecular systems between two different states are governed by two time scales. The Kramers time corresponds to the typical time spent in a given conformation (the dwell time), while the transition path time characterizes the actual duration of the transition. Transition path ti... |
In this paper we have analyzed the TPT distribution of a one dimensional stochastic particle undergoing Langevin dynamics and crossing a parabolic barrier. We investigated the effects of memory and non-equilibrium thus extending previous analysis 10 18 . As the barrier is parabolic, the associated Langevin equation is ... |
In Ref. 19 the effect of memory on transition path times was also investigated. The TPT-distribution was derived for an arbitrary memory kernel starting from an Hamiltonian formulation in which the particle dynamics is coupled to a bath of harmonic oscillators 19 . We expect that the expressions reported in 19 will agr... |
5.1 Long time limit of transition path time distribution: why exponential decay? We have found that the asymptotic decay of the transition path time distribution remains exponential for both cases investigated and therefore has a remarkable universal behavior. This contrasts with the conclusions of Ref. 20 . In that wo... |
To get some more insights about the differences in the effect of memory kernels in the constant force and the parabolic barrier case let us consider the following equation |
[EQUATION] which describes the average motion of particle driven by a constant force [MATH] in a medium characterized by the exponent [MATH] . Using Laplace transforms we find [MATH] , with the normal drift |
[MATH] recovered in the Markovian limit [MATH] This behavior can be deduced from an effective medium description [EQUATION] where the effective friction [MATH] as expected from the time integral of memory kernel, grows with time due to memory effects (the result is consistent with the Einstein relation for the diffusio... |
[EQUATION] with solution [EQUATION] which is a stretched exponential behavior. This is not consistent with the exact solution of the generalized Langevin equation discussed in this paper, which yields for the average position an exponentially growing function at long times (obtained from the asymptotic behavior of the ... |
To understand this apparent paradox, we point out that the effective friction argument would be valid for a process in which the velocity [MATH] is a slowly varying function. For a self-similar process, where the velocity changes according to a power law [MATH] , the coarse grained variable [MATH] by time average behav... |
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