A no-go theorem for ergodicity and an Einstein relation
We provide a simple no-go theorem for ergodicity and the generalized Einstein relation for anomalous diffusion processes. The theorem states that either ergodicity in the sense of equal time and ensemble averaged mean squared displacements (MSD) is broken, and/or the generalized Einstein relation for time averaged diffusivity and mobility is invalid, which is in complete contrast to normal diffusion processes. We also give a general relation for the time averages of drift and MSD for ergodic (in the MSD sense) anomalous diffusion processes, showing that the ratio of these quantities depends on the measurement time. The Lévy walk model is used to exemplify the no-go theorem.
Many processes in disordered systems exhibit anomalous diffusion, that is a nonlinear time dependence of the ensemble average mean squared displacement (MSD) , with for subdiffusion and for superdiffusion (or enhanced diffusion) LWKlaf87 (); LWKlaf90 (); Bou90 (); MetzKlaf00 (). For normal Brownian diffusion, the Einstein relation connects the fluctuations of an ensemble of particles with their mobility (which is the inverse friction) under an applied small constant force ,
where denotes ensemble averaging and and are the particle position with or without applied force , respectively. Eq. (2) holds for normal and anomalous processes close to equilibrium in the limit and can be derived from linear response theory Bou90 (); BarFleu98 (); BeOsh02 (); Zvi09 (). Due to its validity beyond normal processes Eq. (2) is referred to as the generalized Einstein relation (GER). Measurements of this fundamental relation for were performed e.g. for subdiffusive systems by Gu96 (); Amblard96 ().
In what follows we will use the notion of ergodicity in the MSD sense: A process is called ergodic, if time averaged and ensemble averaged MSDs are equal,
where is a time lag (see further discussion below). In the case of Brownian diffusion Eq. (3) holds due to the stationary increments of the process. For certain models of anomalous diffusion ergodicity is violated Lub08 (); HeBur08 ().
In this brief report we formulate a no-go theorem for ergodicity and GER for time averages that applies to all kinds of anomalous diffusions. We then formulate a new type of relation between time averaged drift and MSD for Lévy walks where the ratio of these two quantities is time dependent. We then show that a similar time dependence holds also rather generally for ergodic anomalous diffusion processes. As discussed at the end of this paper, the method of averaging, i.e. time versus ensemble, determines the ratio between fluctuation and drift and hence effective temperatures.
A no-go-theorem for ergodicity breaking and the generalized Einstein relation for time averages.
We define the GER for time averages by substituting ensemble averages by time averages in Eq. (2), that is
In the following we will adhere to the ensemble average of , i.e. the mean taken over many realizations of trajectories so that the respective GER becomes
The time averaged MSD along the trajectory is defined as
where the measurement time is much larger than the lag time . Analogously, the time averaged drift is defined as
With these definitions at hand we assert the general statement that in any system that exhibits anomalous diffusion (sub- or enhanced diffusion) and where a (generalized) Einstein relation of the type Eq. (2) for the ensemble averages holds, at least one of the two properties is violated: either ergodicity in the MSD sense or the GER for time-averages Eq. (5). Thus, let , and , and according to the GER for ensemble averages Eq. (2) , where is a generalized diffusion constant. Then, with Eq. (7) we find
and for in leading order
The GER for the time averages Eq. (5) requires the time averaged MSD
which exhibits a dependence on and if and hence clearly differs from that depends only on one time scale. Ergodicity in the MSD sense is thus broken in this case, . An example for such a process is the subdiffusive continuous time random walk (CTRW) as considered in HeBur08 ().
Conversely, the assumption of ergodicity in the MSD sense clearly implies
This in turn violates the GER for time averages, . To see this, as in the previous case, we use the GER for ensemble averages Eq. (2) which gives Eq. (9). This in turn results in , unless . This case is exemplified by the (Gaussian and ergodic) generalized Langevin systems and fractional Brownian motion Deng09 (). For these ergodic cases where the GER for time averages is violated, we will later discuss the further generalization of the relation between time averaged drift and MSD. The above discussion shows that the GER for time averages and/or ergodicity are broken for anomalous processes. When , both ergodicity and Einstein relation for time averages hold, which constitutes a behavior that is unique to normal diffusion.
Example: Relation of time averaged drift and MSD in the Lévy walk.
We consider a Lévy walk as follows foot (): a particle switches the sign of its velocity at random times. The sojourn times in a velocity state or for which the particle does not change its direction are independent, identically distributed random variables with a common probability density function (PDF) . The initial position at of the particle is . We chose to let the particle start with positive velocity so that it first travels a distance , after that is displaced , then and so forth. The are thereby drawn according to the sojourn time PDF . This sojourn time PDF decays like a power-law at large times, . Depending on the specific choice of this distribution lacks its first moment (), or the second moment (). In our simulations we will use
Thus in the unbiased case (the biased case will be discussed below), the large-time asymptotics reads for and for in Laplace domain, with the Laplace variable conjugate to . Here , and .
The ensemble averaged mean squared displacement for this process is well known ZumKlaf90 () and yields for
a quadratic time dependence so that we refer to this regime as the ballistic one. For we have
whose time dependence is slower than quadratic, but faster than linear so that we call this regime subballistic (or enhanced diffusion regime). Here we introduced the generalized diffusion coefficient, remark ().
For this type of Lévy walk, depending on the time averaged MSD Eq. (6) can be a random quantity. The properties of the fluctuations of were investigated recently ours (); ours2 (). Its ensemble average yields in particular
i.e. differs from the respective by a factor ours (); ours2 (); Akimoto12 (); Ralf13 (). Note however that in the subballistic case the ensemble averaged MSD depends crucially on the initial conditions: For the equilibrated process that started long before the beginning of the measurement we find Akimoto12 (); ours (); ours2 ()
where denotes the average over an ensemble of stationary processes (in contrast to the average over an ensemble conditioned on a turning event taking place at ). Thus, based on the equilibrium MSD one would conclude that the subballistic Lévy walk is ergodic. For the average sojourn time is infinite. Hence an equilibrated state does not exist so that strictly speaking weak ergodicity breaking B92 (); MarBar05 () occurs only in the ballistic case.
Let us further consider a small constant force acting on the particle of mass . At each renewal event, the particle starts out at the respective velocity as given in the force-free case discussed above, but then accelerates according to Newton’s law for the duration of the waiting times drawn from the PDF Eq. (12). This setting leads to a net time averaged drift described by Eq. (7) involving an integral over the ensemble averaged drift . The ensemble averaged drift was calculated earlier BarFleu98 (). In our particular case we have
which for the subballistic case holds also for the drift under equilibrium initial conditions, . The GER for ensemble averages Eq. (2) holds also for the Lévy walk, as is easily verified with Eqs. (13), (14) and (17): , where we can assign an effective kinetic temperature remark1 ().
time averaged GER violated
time averaged GER valid
In normal Brownian diffusion, the Einstein relation (2) holds also for the time averages, a trivial observation following from ergodicity, i.e. equality of time and ensemble averages for long observation times. For the Lévy walk, using Eqs. (7) and (17), the ensemble averages of the time averaged drift yield
These results differ from their corresponding ensemble average . In the ballistic case , we have a linear instead of a quadratic dependence on the time lag. In the subballistic case the dependence on the lag is also linear, as compared to . In both cases we find a dependence on two time scales, and . In particular, the time averaged response to an external bias is increasing with . Hence, in the ballistic regime and in the subballistic regime under nonequilibrium initial conditions the GER Eq. (5) and ergodicity are violated, compare Eqs. (18) and (15). For equilibrium preparation, the subballistic case is ergodic (see Eq. (16)), but the time averaged GER Eq. (5) is violated.
with in the ballistic and in the subballistic case. This establishes the relationship between dispersion and drift in the present model. The limiting case where the sojourn times in the velocity states possess first and second moment re-establishes the normal diffusion case, so that after sufficiently many collisions or changes of direction we have a stationary state. Hence in this limit from Eq. (19) the well known Einstein relation Eq. (5) is recovered for the time averages (where we assign ). Generally, however, care has to be taken with interpretation of relation Eq. (19) in the sense of a fluctuation-dissipation relation, since does not necessarily only comprise thermal contributions. Conversely, the long excursions leading to ballistic or subballistic anomalous transport may require nonthermal energy input that keeps the system out of equilibrium, as is put into effect e.g. by motor proteins in subballistic transport within living cells Caspi02 (). This being said, the question of the generality of such a time dependent drift-fluctuation ratio Eq. (19) arises. We will come back to that later.
Figs. 1, 2 show the numerically obtained ratio of the ensemble average drift under the small force , and the mean of the time averaged MSD, . The numerics corroborate the theoretical predictions Eq. (19) quite well, although in the subballistic case the convergence was quite poor, especially for smaller . Thus the numerical results shown in Figs. 1, 2 also rule out the relation Eq. (5) for the Lévy walk.
We define an effective mobility via and find in the ballistic phase , while in the subballistic case . Thus the effective mobility increases with the total observation time: linearly in the first and sublinearly in the latter case remark3 ().
Coming back to a more general setting and to the question of how exactly, according to our no-go theorem, the GER for time averages is violated if the process is ergodic, we now make a stronger assumption: We suppose that the anomalous diffusion process is ergodic in the MSD sense and that the GER for ensemble averages Eq. (2) holds. Thus, Eq. (9) follows immediately from the anomaly, and ergodicity implies , and . Hence in this case
which resembles Eq. (19) up to a factor . In fact, in the subballistic phase of the Lévy walk this factor accounts for the nonequilibrium initial preparation of the system ours2 (); Akimoto12 () which renders the system nonergodic in the MSD sense. Eq. (20) thus holds for the ergodic Lévy walks under equilibrium preparation.
We provided a rather general no-go theorem for the validity of ergodicity and the GER for time averages: At least one of the two properties is violated in systems exhibiting anomalous sub- or superdiffusion, see Table 1. This no-go theorem is based on the two assumptions that the process exhibits anomalous diffusion and that the system obeys the GER for the ensemble averages which leads to , . For anomalous diffusion processes the average response to an exerted force exhibits clearly a different time dependence than , which is due to the non-linear time dependence of .
In particular, we investigated the Lévy walk model. While the GER in the Lévy walk holds for ensemble averages, it is violated for the time averages. This is in sharp contrast to the subdiffusive CTRW system studied in HeBur08 () where ergodicity is violated but a time average GER holds. The time dependent ratio of drift and dispersion in Eq. (19) entails a mobility effectively increasing (or a decreasing friction) with the measurement time, reflecting the active character of the anomalously large excursions in the Lévy walk. The Lévy walk constitutes therefore an example for a system where the relation between time averaged drift and MSD differs considerably from that of mere ensemble averages. Moreover we have derived a general relation between time averaged drift and MSD for ergodic anomalous diffusion processes, Eq. (20).
Many works are devoted to the question on the ratios between fluctuations and drift which are used to define effective temperatures Cugl11 (); FieSol02 (); Barr99 (). In our case the temperature is well defined, however the ratio is time dependent. Depending on the averaging procedure, either time or ensemble averaging, we get different effective temperatures for many kinds of anomalous processes. Thus when discussing the ratios of fluctuations and drift, the method of averaging has to be specified carefully. Theories so far focused on ensemble averages, while experiments seem to focus on time averaging (or a mixture of both).
Acknowledgement: This work was supported by the Israel Science Foundation.
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