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. Thus both beat plane orientation and the specifics of boundary approach are dependent upon virtual promastigote morphology, and together result in the flagellar tip being the primary point of surface contact. |
3.4 Swimming is unstable in the absence of a surface force As described in Section 2.6 , we compute a phase plane in order to deduce long-time behaviour (see Fig. |
6(a) ). Partitioning phase plane trajectories by the [MATH] nullcline (which approaches [MATH] in the far-field, depicted blue, dashed), we observe that almost all virtual promastigotes approaching the boundary from the far-field will eventually collide with the boundary (see Supplementary Movie 2). Similarly, almost a... |
initially increases and then approaches a constant value as promastigote-wall separation grows. However, the phase plane shows a region where non-monotonic change in [MATH] |
appears plausible (see Fig. 6(b) ), suggesting that those promastigotes beginning on trajectories in the region between the [MATH] and |
[MATH] -nullclines will initially move away from the boundary, but will subsequently undergo reorientation towards the surface and eventually collide with it. Contrastingly, such behaviours are not observed in full simulations, however this is not a significant conflict given the magnitude of the error introduced by th... |
initially, which orient away from the boundary rather than swimming stably (see Supplementary Movie 3). In order to compare these behaviours against an appropriate pusher, we reverse the direction of beat propagation in the flagellum and simulate the resulting motion. The behaviour of this pusher is captured by a phase... |
Fig. ), where we observe that [MATH] is a stable attractor of the system, corresponding to swimming parallel to the boundary. In long-time simulations of the full system we in fact observe stable boundary swimming at a constant separation [MATH] , which differs slightly from the beat-averaged system due to the previous... |
, subject to a phase shift. As time appears only as a parameter in the governing equations, the solution to the time-reversed problem is simply the reversal of the original problem. Therefore we recover the reversed behaviour of the virtual promastigote in the behaviours of this pusher, and hence observe stable paralle... |
3.5 Repulsive surface forces do not give rise to stable boundary swimming The repulsive surface potential of Section 2.5 is introduced to the system. Due to the short range over which the resulting repulsive force is non-negligible, the previous phase plane analysis that does not account for the surface force holds thr... |
for an example) to ascertain the motion of the virtual promastigote when in close proximity to the boundary, following which the phase plane of Fig. |
6(a) is used to examine further behaviour. One might envisage the existence of a periodic motion, with the torque induced by the surface potential orienting the cell into a configuration where it will again collide with the boundary, and such motion repeats ad infinitum. This phenomenon could not be observed for all co... |
We observe that the only exhibited behaviour is deflection away from the boundary, with the surface repulsive force causing reorientation of the promastigote such that it falls into the deflective regime, which is observed to be unchanged for reasonable variation in force strength due to the short-range nature of the s... |
Reversing the direction of beat propagation we examine the near-boundary behaviour of the morphologically-equivalent pusher, which due to the repulsive surface force may not be inferred by time reversal of the puller behaviour. In contrast to the deflection observed for the puller, in this case we in fact see that the ... |
Discussion and conclusions In this work we have identified and described a novel model flagellar waveform of L. mexicana promastigotes, observing a simple planar dynamics in [MATH] cells that lends itself to simple parameterisation and subsequent simulation. Using this model beat pattern we have observed and explored t... |
We have seen that swimming near a boundary in the absence of surface forces is unstable for virtual promastigotes, with trajectories resulting either in immediate deflection away from the boundary or eventual collision with the surface. This behaviour may not be deduced from previous observations of puller microswimmer... |
. Here, promastigotes that initially swim away from a boundary may be captured if sufficiently close, undergoing reorientation which results in their collision with the surface. Drawing comparison with in vivo promastigotes, this may provide hydrodynamic explanation for the epithelial attachment observed in the sandfly... |
Further, we have noted that time-reversibility of the governing equations allows us to immediately compare the dynamics of pushers and pullers near boundaries in scenarios without repulsive surface forces. Whilst our results have shown that no stable boundary swimming occurs in the case of our virtual promastigote pull... |
. A significant reduction in cell body size is not observed to drastically alter the behaviour of the virtual pusher, with only the stable swimming height being affected. Thus our results show that the stable boundary swimming of monoflagellates in quiescent fluid is highly dependent on the method of locomotion, and le... |
We have also seen that a change in surface character, from a scenario in the absence of surface forces to one with a short range surface force, reduces the prevalence of surface-bound trajectories, with those that would previously have ended in collision now being reoriented into the deflective regime. This promotion o... |
Indeed, we conjecture that this taxis is aided by the transfer of promastigotes into the bulk and away from the no-slip boundary, so that cells are more susceptible to convection by background flows. Such flows may be associated with sandfly regurgitation, stimulated by the parasite’s production of promastigote secreto... |
. Thus the hydrodynamic interaction behind the deflection of virtual promastigotes may be responsible for in vivo movement of promastigotes into the bulk, and could therefore facilitate the transmission of the infective form of the parasite from vector to host. |
However, whilst we have noted that stable boundary swimming of virtual promastigotes does not occur with a repulsive surface potential, this is not the case for the human spermatozoon |
. Differing in both hydrodynamic classification and cell morphology, it is not clear in the context of the presented results which feature drives the contrasting behaviours observed when accounting for short range surface forces, as time-reversibility is lost due to the boundary force. However, the above results indica... |
Remarkably, we have observed a morphology-dependent mechanism for the promotion of tip-first boundary collision for virtual promastigotes near non-repulsive boundaries. We hypothesise that this mechanism may provide an explanation for the observed behaviour of in vivo promastigotes, where flagellum-first attachment is ... |
. Here in silico , our study of virtual promastigotes suggests a refined mechanism, whereby the comparatively large body size of the promastigote results in the emergence of drag-based reorientation of notable magnitude, highly dependent on body lengthscale, aligning the virtual flagellum such that the distal tip initi... |
In Appendix we have examined the behavioural effects of changing body lengthscale and aspect ratio, demonstrating that reported behaviours are robust to typical observed variation in these morphological parameters. There remains significant scope in future work to relax the assumption of body axisymmetry in order to mo... |
In summary, we have investigated in detail the boundary behaviours of a flagellated puller, a virtual L. mexicana promastigote equipped with a determined planar tip-to-base beat pattern, and have observed that stable boundary accumulation does not feature amongst the range of exhibited behaviours, irrespective of the i... |
Acknowledgements B.J.W. is supported by the UK Engineering and Physical Sciences Research Council (EPSRC), grant EP/N509711/1. R.J.W. is supported by a Wellcome Trust Sir Henry Wellcome Fellowship [103261/Z/13/Z] and a Wellcome Trust Sir Henry Dale Fellowship [211075/Z/18/Z], with equipment supported by a Wellcome Trus... |
Appendix A Incompressible Stokes equations Briefly stated, the dimensional Stokes equations for an incompressible Newtonian fluid with velocity field [MATH] and pressure [MATH] are given by |
[EQUATION] where [MATH] is the dynamic viscosity of the fluid. We non-dimensionalise with lengthscale [MATH] and velocity scale [MATH] , and scale pressure with [MATH] , taking the advective timescale [MATH] and using the dynamic viscosity of water at 25 [MATH] . For typical L. mexicana values this gives a Reynolds num... |
. This equation is then applied to the surface of the virtual promastigote and discretised, with the resulting linear system being solved for cell velocities |
[MATH] given the prescribed flagellar kinematics, together with the force and torque conditions on the swimmer. Appendix B Meshing the virtual promastigote |
Both icosahedral and triangulated-cubic meshes were used for the discretisation of the virtual promastigote body. Subdivision was performed by the bisection of existing edges, increasing the element count by a factor of 4 per subdivision. For the flagellum, a cylindrical mesh with spherically-capped ends was used, with... |
Appendix C Effects of body morphology Given the reported variation in Leishmania promastigotes , we examine the effects on swimming of altering the morphological parameters describing the swimmer body. With the body length of typical promastigotes varying approximately between 7 [MATH] and 17 [MATH] |
, we simulate the motion of modified virtual promastigotes with body lengths sampled from this range. Here we fix the body aspect ratio to be that of the virtual promastigote, and show sample kinematics in Fig. |
for a single initial configuration. Fig. 1(a) presents the motion of swimmers when a repulsive boundary force is included. We observe that, despite differing in body lengthscale, the swimmers follow qualitatively-similar paths in phase space and exhibit the same overall behaviours. The larger-bodied swimmers are seen t... |
. Without repulsive short range surface forces the same level of qualitative agreement is present, as evident from the time series of Fig. |
1(b) , with increased reorientation towards the boundary for larger-bodied swimmers. Similar consideration of variation in body aspect ratio from that of the virtual promastigote yields qualitatively-unchanged dynamics, where additionally we observe that decreases in aspect ratio result in reduced swimmer velocities an... |
. Hence we have seen that the reported behaviours of virtual promastigotes are robust to oberved variations in both aspect ratio and body lengthscale. References |
# Source: arxiv 1806.00435 # Title: Emergence of correlations in the process of thermalization of interacting bosons # Sections: all # Downloaded: 2026-03-02T08:51:03.789221+00:00 |
Emergence of correlations in the process of thermalization of interacting bosons Abstract We address the question of the relevance of thermalization to the increase of correlations in the quench dynamics of an isolated system with a finite number of interacting bosons. Specifically, we study how, in the process of ther... |
pacs: 05.30.-d, 05.45.Mt, 67.85.-d Introduction - In recent years the problem of thermalization in closed systems of interacting fermions and bosons has attracted much attention (see, for example, Refs. reviews BISZ16 ). An increase of interest to this problem is due to remarkable experimental achievements exp and vari... |
To date it is understood that the validity of statistical mechanics can be justified not only by averaging over a number of eigenstates with close energies, but also with the use of a single eigenstate if the latter consists of many uncorrelated components in the physically chosen basis. Specifically, it was shown that... |
Unlike the onset of BE and FD distributions emerging from single stationary eigenstates, in this Letter we address a new problem concerning the onset of the BE distribution in the evolution of a system with few interacting bosons. Our specific interest is to study how the conventional BE distribution emerges in time an... |
In our study we consider the quench dynamics described by the Hamiltonian [MATH] where [MATH] represents the non-interacting bosons and the interaction is fully embedded into [MATH] belonging to the ensemble of two-body random interacting (TBRI) matrices. In this way, by exciting initially a single many-body state of [... |
Below, in connection with the results reported in BMI17 BIS18 we show, both analytically and numerically, that the onset of the BE distribution in the TBRI matrix model occurs on the time scale on which the number of components in many-body eigenstates increases exponentially in time. In order to quantify the onset of ... |
The model - The system consists of [MATH] identical bosons occupying [MATH] single-particle levels specified by random energies [MATH] with mean spacing, [MATH] . The Hamiltonian [MATH] reads ( [MATH] ), |
[EQUATION] where the two-body matrix elements [MATH] are random Gaussian entries with zero mean and variance [MATH] . The dimension of the Hilbert space [MATH] Here we consider [MATH] particles in [MATH] levels (dilute limit, [MATH] ) for which [MATH] . Two-body random matrices ( ) were introduced in TBRI brody and ext... |
The eigenstates [MATH] of [MATH] can be written in terms of the basis states [MATH] of [MATH] , where [EQUATION] An eigenstate [MATH] of the total Hamiltonian is called chaotic when its number [MATH] of principal components |
[MATH] is sufficiently large and [MATH] can be considered as random and non-correlated ones. Note that since the system is isolated and the perturbation [MATH] is finite, the eigenstates can fill only a part of the unperturbed basis BISZ16 determined by the perturbation [MATH] . Specifically, the energy region which is... |
Dynamics in Fock space - In contrast with the previous studies BMI17 , focused on the thermal properties of individual many-body eigenstates, here we consider the dynamics of the model ( ) by exploring two different time scales, before and after the relaxation to a steady state. Specifically, we study the quench dynami... |
[MATH] one can express the probability [MATH] to find the system at time [MATH] in any unperturbed state [MATH] as follows, [EQUATION] |
where [MATH] and [MATH] are the time-independent and time-fluctuating parts, respectively. With this expression, one can analyze the number of principal components, |
[EQUATION] known as the participation ratio. Taking the long-time average, [MATH] cancels out and only the diagonal part [MATH] survives. As is shown in BIS18 the number [MATH] of principal components in the wave packets increases exponentially fast in time, [MATH] up to some saturation time [MATH] . The rate of the ex... |
[EQUATION] obtained by projecting the initial state [MATH] onto the energy eigenstates. In nuclear physics it is known as strength function and it describes the relaxation of excited heavy nuclei bohr Concerning the saturation time [MATH] , it was found BIS18 to be proportional to the number of particles, [MATH] . This... |
Onset of Bose-Einstein distribution - The time-dependent occupation number distribution (OND) is defined as follows, [EQUATION] It gives the average number of particles in the single-particle energy level [MATH] at the time [MATH] . Here we took into account that |
[MATH] where [MATH] . The evolution of [MATH] in comparison with the wave packet dynamics [MATH] is shown in Fig (e)-(h). This figure demonstrates that when the packet fully occupies the energy shell, the occupation numbers are relaxed to the steady-state distribution. |
Expanding [MATH] at second order one gets the time dependence for [MATH] at small times, [EQUATION] which results in the following estimate, |
[EQUATION] One can see in Fig. that for single-particle [MATH] levels which are not initially occupied by particles, [MATH] grows quadratically in time. As for the saturation values [MATH] after the relaxation time [MATH] , they can be also obtained analytically by performing an infinite time average, |
[EQUATION] In order to claim that after relaxation the OND is statistically described by a BE distribution, one has to be sure that the fluctuations of [MATH] follow the standard requirements of statistical mechanics. In view of this very point, we have thoroughly analyzed both “classical” and “quantum” fluctuations. C... |
[MATH] in the wave packet plays the same role as the number of particles [MATH] in ordinary statistical mechanics. A more intriguing point concerns quantum fluctuations. It is a textbook result huang that BE statistics is characterized by relative quantum fluctuations |
[MATH] , where [MATH] with the overbar standing for the infinite time-average, see Eq. ( ). Once again we checked that, provided the time-dependent wave function is chaotic, fluctuations follow the predictions of standard statistical mechanics (for details see SM ). This should be considered as an additional proof of t... |
Two-point correlation function - Let us now study how the onset of the BE distribution is manifested by the emergence of correlations between occupation numbers. First, we start with the two-point correlation function [MATH] between neighboring occupation numbers, |
[EQUATION] Initially the correlations are absent, [MATH] , however, they appear in time. The time-dependence of [MATH] is shown in Fig. for all [MATH] . As one can see, there is a clear relaxation to steady-state values after the critical time [MATH] . The negative or positive sign of the asymptotic correlations is rel... |
It is also instructive to introduce the global correlator [MATH] which is the sum of the correlators between all neighboring single-particle energy levels [MATH] and [MATH] |
[EQUATION] This correlator is independent of the specific [MATH] level and it can be used as a global measure of correlations between occupation numbers of nearest single-particle energy levels. Performing an expansion on a small time scale it is possible to show that |
[EQUATION] with [MATH] As one can see, Eq. ( 11 ) does not contain eigenvalues and eigenfunctions. This means that in order to get the initial spread of the correlator, there is no need to diagonalize the Hamiltonian. Concerning the saturation value, it can be obtained by performing the time average for [MATH] (see SM ... |
[EQUATION] The time evolution for [MATH] is shown in Fig. , together with the analytical predictions. The correspondence between numerical data and analytical predictions is impressive. Thus, the dynamics of [MATH] is fully described by the analytical expressions ( 11 ) and ( 12 ). |
Four-point correlation function (OTOC) - Now let us study the four-point correlator between nearest single-particle energy levels, |
[EQUATION] This correlator, also known as OTOC, has been recently introduced in the frame of the SYK model SYK and widely discussed in view of various physical applications (see e.g. OTOC ). |
After some algebra SM , one can obtain that the correlator [MATH] increases in time quadratically on a small time scale, whose validity defines the perturbative regime, |
[EQUATION] In the same way, by performing an infinite time-average, we can obtain the steady state value [MATH] [EQUATION] with [MATH] |
Numerical data for [MATH] are shown in Fig. together with the expressions ( 14 ) and ( 15 ). Our results demonstrate that while in the perturbative regime the growth is indeed quadratic, a time window can be found where the correlator increases approximately as [MATH] , before the saturation. This occurs at variance wi... |
Conclusion and discussion - In this Letter we address the question of how the conventional Bose-Einstein distribution emerges in an isolated system with a finite number of interacting bosons. Since this process is accompanied by an increase of strong correlations between occupation numbers [MATH] , the large part of ou... |
For our analysis we have used the well known model ( ) describing bosons interacting to each other via two-body random matrix elements. By exploring the quench dynamics, we show that the BE distribution emerges on the same time scale [MATH] |
on which the number of principal components in the wave function increases exponentially in time in the Fock space BIS18 This time scale [MATH] is proportional to the number [MATH] of bosons and defines the time after which one can speak of a complete thermalization in the system. |
In order to confirm the true statistical behavior of the occupation numbers, we have carefully studied the fluctuations of [MATH] after the relaxation. In accordance with the standard statistical mechanics our data manifest that the fluctuations are of the Gaussian type, and that they are small compared to the mean val... |
In order to reveal how the process of thermalization is related to the onset of correlations, we have studied, both analytically and numerically, two correlators. One is the standard two-point correlator between nearest occupation numbers [MATH] and [MATH] and the other is the out-of-time order correlator (OTOC) recent... |
Our results show how the information initially encoded in a local unperturbed state, spreads over the whole system and transforms onto global correlations specified by the BE distribution of occupation numbers. Although the dynamics is completely reversible due to the unitarity of the evolution operator, it is practica... |
We hope that our study can help to understand the relation between thermalization and scrambling from one side, and the onset of correlations in the evolution of chaotic systems from the other one. Since the TBRI matrix model ( ) has been proved to manifest generic statistical properties occurring in realistic physical... |
Acknowledgements. We acknowledge financial support from VIEP-BUAP Grant No. IZF-EXC16-G (FMI) and Iniziativa Specifica INFN-DynSysMath (FB). |
Supplemental Material for : Emergence of correlations in the process of thermalization of interacting bosons Fausto Borgonovi Felix M. Izrailev |
Supplemental Material: Emergence of correlations in the process of thermalization of interacting bosons Fausto Borgonovi 1,2 , Felix M. Izrailev 3,4 |
Dipartimento di Matematica e Fisica and Interdisciplinary Laboratories for Advanced Materials Physics, Università Cattolica, via Musei 41, 25121 Brescia, Italy |
Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, via Bassi 6, I-27100, Pavia, Italy Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico |
Dept. of Physics and Astronomy, Michigan State University, E. Lansing, Michigan 48824-1321, USA Dynamics Let us consider initially an unperturbed many-body state of [MATH] |
[EQUATION] whose evolution under the Hamiltonian [MATH] is given by [EQUATION] (note that all [MATH] are real numbers). The probability to be in the unperturbed many-body state [MATH] is |
[EQUATION] which can be written as a diagonal (time independent) plus a fluctuating (time-dependent) part, [EQUATION] Let us now define the long-time average of an observable [MATH] as |
[EQUATION] It is clear that for a non-degenerate spectrum [MATH] so that, [EQUATION] I.1 Number of Principal Components The long-time average for the number of principal components can be computed as follows. Let us start from its definition, |
[EQUATION] Taking the infinite-time average we have [EQUATION] The second term in the r.h.s. of Eq. ( 23 ) can be computed exactly, |
[EQUATION] so that the long-time average for the number of principal components is given by, [EQUATION] This expression determines the asymptotic value reached by [MATH] after relaxation. It is shown in Fig. (a) as a horizontal line. In the same figure we can identify three different regimes : a perturbative one for sh... |
[MATH] grows quadratically (see inset in Fig. (a)); a second one characterized by the exponential growth, [MATH] for [MATH] , and a third one (saturation after relaxation) where [MATH] for [MATH] (for details see bis18 ). |
Another important information is how the stationary value [MATH] depends on the initial state. In Fig. (b) we show [MATH] as a function of the unperturbed energy [MATH] of the initial many-body state [MATH] As one can see it is quite well approximated (excluding the tails) by a Gaussian shape (see black full curve). |
I.2 Single-particle Occupation Numbers Time dependent single-particle occupation numbers are defined as, [EQUATION] Performing the infinite time average one obtains for the first two moments, |
[EQUATION] and from that [EQUATION] where the dependence on [MATH] has been explicitly indicated in Eq. ( 28 ). I.3 Two-point Correlation Function |
First of all let us notice that the number operator [MATH] giving the number of particles in the single-particle energy level [MATH] is diagonal in the unperturbed many-body basis, i.e. |
[EQUATION] Concerning the global two-point correlation function one has, starting from the initial state [MATH] [EQUATION] where [MATH] . In Eq. ( 30 we have defined |
[EQUATION] The long-time average is thus given by, [EQUATION] I.4 Four-point Correlation Function Let us obtain the long-time estimate for the four-point correlation function (OTOC): |
[EQUATION] From the definition it is clear that [MATH] In order to compute explicitly Eq. ( 33 ) let us insert a completeness so that, |
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