Com relação as perspectivas sobre o modelo de memória exponencial, o nosso pensamento é desenvolver a solução analítica para encontrar o conjunto de valores para o expoente de Hurst. Já iniciamos esse procedimento na seção (5.4). Utilizamos como ponto de partida a notação usada pelo trabalho [74]. Conseguimos encontrar duas equações, uma delas se refere ao termo periódico, frequência B, equação (5.4.30) e a outra equação (5.4.31) é a relação para o expoente δ. O expoente δ irá fornecer os possíveis valores para o expoente de Hurst. A equação para o parâmetro B está relacionado com a caracterís- tica da log-periodicidade que alguns modelos com memória apresentaram no feedback negativo, região p < 1/2, ou seja, o expoente de Hurst é H > 1/2. Os nossos resulta- dos numéricos do modelo com memória exponencial não apresentaram a característica de log-periodicidade. O resultado analítico encontrado, equação (5.4.33), pode fornecer uma estimativa para os valores do expoente de Hurst, na região onde B = 0. Nessa região não existe log-periodicidade. Lembrando que toda a análise analítica feita na seção (5.4) precisa ser verificada com mais profundidade.
Uma outra perspectiva para o modelo com memória exponencial seria traba- lhar uma abordagem estatística para modelo. Vamos aplicar as ideias da estatística não- extensiva de Constantino Tsallis [79, 80]. O ponto de partida da abordagem estatística seria calcular a entropia generalizada
Sq[p] = {1 −
Z
[p(x)]qdx}/(q − 1). (6.2.1)
Maximizando a entropia com a imposição de que o vínculo seja finito Z
x2[p(x)]2dx (6.2.2)
e também com a exigência de Z
p(x)dx = 1. (6.2.3)
Notamos que para calcular a expressão da entropia é essencial conhecer a distribuição p(x) que maximiza a entropia sujeita aos vínculos. Um vez encontrada a relação para a entro- pia, podemos explorar parâmetros macroscópicos relacionados ao modelo com memória exponencial. É ensinado no curso de mecânica estatística que a entropia é uma grandeza que conecta as propriedades termodinâmicas com um ensemble. Portanto, obtendo a rela-
ção para a entropia generalizada, podemos investigar as propriedaddes termodinâmicas do modelo com memória exponencial.
A última perspectiva seria trabalharmos a função distribuição de probabilidade invertida P (t′
) ∝ eλ(t−t′)t . Esperamos que essa função distribuição de probabilidade apre- sente superdifusão log-periódica.
[1] Richard P. Feynman, Robert B. Leighton and Matthew Sands The Feynman Lectures on Physics, Bookman Companhia Editora Ltda, vol. I cap. 37 (2008).
[2] Silvio R. A. Salinas, Introdução a Física Estatística, (Editora, EdUSP, 2ª Ed., ISBN 8531403863), (2005).
[3] Antonio de Macedo Filho,Processo de Difusão com Agregação e Reorganização Espontânea em uma Rede 2d, Natal-RN (2008), Dissertação (Mestrado), Departamento de Física Teórica e Experimental - UFRN.
[4] http://proascg2.pbworks.com/f/FOLHA.jpg [5] http://imaginacaocriativa.blogspot.com.br/ [6] http://www.pt.wikipedia.org/wiki
[7] C. A. A. Araújo, E. L. Albuquerque, D. H. A. L. Anselmo, M. S. Vasconcelos, Magnetic Polaritons in Metamagnet Layered Structures: Spectra and Localization Properties, Physics Letters A 372, 1135-1140 (2008).
[8] L. S. Lucena, J. M. Araújo, D. M. Tavares, L. R. da Silva and C. Tsallis, Ramified poly- merization in dirty media: A new critical phenomenon, Phys. Rev. Lett. 72, 230-233 (1994). [9] B. B. Mandelbrot, Fractals: Form, Chance and Dimension, San Francisco: Freeman,
(1977).
[10] R. M. Ziff., P. T. Cummings, G. Stell, J. Phys. A, Generation of percolation cluster perime- ters by a random walk 17, 3009 (1984).
[11] R. Brown, A brief account of microscopical observations made in the months of june, july, and august, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies, Philosophical Magazine N. S. 4, 161173, (1828).
[12] Angel Akio Tateishi, Desenvolvimento do Conceito de Difusão: De Fourier ao Modelo de Pente, Paraná (2010), Dissertação (Mestrado) - Departamento de Física - Universidade Estadual de Maringá.
[13] Silvio R. A. Salinas, Einstein e a teoria do movimento browniano, Revista Brasileira de Ensino de Física 27, 263-269 (2005).
[14] Tânia Tomé, Mário J. de Oliveira, Dinâmica Estocástica e Irreversibilidade (Editora EdUSP., 1ª Ed. ISBN 8531405955), (2001).
[15] Linda E. Reichl, A modern Course in Statistical Physics, (Published, John Wiley Profes- sio, 2ª Ed.), (1998).
[16] N.G. van Kampen, Remarks on Non-Markov Processes, Brazilian Journal of Physics 28, (1998).
[17] Marcelo Felisberto de Lima, Processos Estocásticos não-Markovianos em Difusão Anô- mala, Tese (Doutorado), Instituto de Física - UFAL (2011).
[18] L. F. Richardson, Mathematical psychology of war, Nature 135, 830?831 (1935).
[19] L. F. Richardson, Some measurements of atmospheric turbulence, Philos. Trans. R. Soc. London Ser. A 221, 128 (1920).
[20] L. F. Richardson, Atmospheric Diffusion Shown on a Distance-Neighbour Graph, Proc. R. Soc. Lond. A 110, 709?737 (1926).
[21] D. L. Styris, C. T. Tomikuza, Anomalous Diffusion Rate for Small Penetration Distance in Copper, J. Appl. Phys. 34, 1001 (1963).
[22] K. H. Nicholas, Studies of Anomalous Diffusion of Impurities in Silics, Solid-State Elec- tron. 9, 35-47 (1966).
[23] H. Scher and E. W. Montroll, Anomalous transit-dispersion in amorphous solids, Phys. Rev. B 12, 2455-2477 (1975).
[24] E. W. Montroll and H. Scher, Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries, J. Stat. Phys. 9, 101135 (1973).
[25] Ismael Victor de Lucena Costa, Processos estocásticos: difusão e crescimento, Brasília (2006), Tese (Doutorado), Instituto de Física - Universidade de Brasília.
[26] H. Mori, Transport collective motion and brownian motion, Prog. Theor. Phys. 33, 423 (1965).
[27] H. Mori, A continued-fraction representation of time-correlation functions, Prog. Theor. Phys. 34, 399 (1965).
[28] M. H. Lee, Derivation of the generalized Langevin equation by a method of recurrence rela- tions, J. Math. Phys. 24, 2512 (1983).
[29] M. H. Lee, Generalized Langevin equation and recurrence relations, Phys. Rev. E 62, 1769 (2000).
[30] Rafael Morgado Silva, Difusão e tempo de escapamento para um sistema com memória de longo alcance, Brasília-DF (2001), Dissertação (Mestrado), Instituto de Física da Uni- versidade de Brasília.
[31] F. A. Oliveira, R. Morgado, C. Dias, G. G. Batrouni and A. Hansen, Comment on Nons- tationarity Induced by Long-Time Noise Correlations in the Langevin Equation?, Phys. Rev. Lett. 86, 5839 (2001).
[32] R. Morgado, F. A. Oliveira, G. G. Batrouni, Alex Hansen, Relation between anoumalous and normal diffusion in systems with memory, Phys. Rev. Lett. 89, 100601 (2002).
[33] R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II, Springer, (1991).
[34] Douglas Azevedo Sant´Anna, Derivadas Fracionárias, Funções Contínuas Não- Diferenciáveis e Dimensões, São Paulo (2009), Dissertação (Mestrado), Programa de pós-graduação em Matemática, Universidade Federal do ABC.
[35] C. K. Peng, J. Mietus, J. M. Hausdorff, S. Havlin, H. E. Stanley, and A. L. Goldberger, Long-range anticorrelations and non-Gaussian behavior of the heartbeat, Phys. Rev. Lett. 70, 1343-1347 (1993).
[36] Vasiliki Plerou, Parameswaran Gopikrishnan, Bernd Rosenow, Luís A. Nunes Ama- ral, Thomas Guhr, and H. Eugene Stanley, Random matrix approach to cross correlations in financial data, Phys. Rev. Lett. 62, 3023-3026 (2000).
[37] J. Klafter, M.F. Schlesinger, G. Zumofen, Beyond Brownian motion, Phys. Today, New York, 49, 33-39 (1996).
[38] R. Hilfer, Applications of Fractional Calculus in Physics, Singapore: World Scientific, (2000).
[39] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Rep. 339, 1-77 (2000).
[40] Ralf Metzler, Eli Barkai, Joseph Klafter, Anomalous transport in disordered systems under the influence of external fields, Physica A 266, 343-350 (1999).
[41] G. Gonçalves et al., Difusão anômala e equações fracionárias de difusão, Acta Sci. Technol. 27, (2005).
[42] S.S. Plotkin and P.G. Wolynes, Non-Markovian Configurational Diffusion and Reaction Coordinates for Protein Folding, Phys. Rev. Lett. 80, 5015-5018 (1998).
[43] Constantino Tsallis, RenioS. Mendes, A.R. Plastinoa, The role of constraints within ge- neralized nonextensive statistics, Physica A 261, 534-554 (1998).
[44] D. Prato and C. Tsallis, Nonextensive foundation of Lévy distributions, Phys. Rev. E 60, 2398-2401 (1999).
[45] M.F. Schlesinger, G.M. Zaslavsky and U. Frisch, Lévy Flights and related topics in phy- sics, Berlin: Springer-Verlag, (1994).
[46] K.B. Oldham and J. Spanier, The Fractional Calculus, New York: Academic Press, (1974).
[47] Loukas Vlahos, Heinz Isliker, Yannis Kominis and Kyriakos Hizanidis, Normal and Anomalous Diffusion: A Tutorial, Preprint submitted to Elsevier (2008).
[48] E.W. Montroll and G.H. Weiss, Random Walks on Lattices. II, J. Math. Phys. 6, 167 (1965).
[49] M.F. Shlesinger, B. West and J. Klafter, Lévy dynamics of enhanced diffusion: Application to turbulence, Phys. Rev. Lett. 58, 1100 (1987).
[50] G. Zumofen, J. Klafter, Scale-invariant motion in intermittent chaotic systems, Phys. Rev. E 47, 851 (1993).
[51] Marcos Cesar Santos, Busca Markoviana Super Difusiva: Origens e Relevância das Leis de Potência, Curitiba-PR (2008), Dissertação (Mestrado), Pós-Graduação em Física da Universidade Federal do Paraná.
[52] Gandhimohan M. Viswanathan, Fish in Lévy-flight foraging, Nature 465, 1018-1019 (2010).
[53] Andrew M. Edwards, Richard A. Phillips, Nicholas W. Watkins, Mervyn P. Freeman, Eugene J. Murphy, Vsevolod Afanasyev, Sergey V. Buldyrev, M. G. E. da Luz, E. P. Raposo, H. Eugene Stanley and Gandhimohan M. Viswanathan, Revisiting Levy flight search patterns of wandering albatrosses, bumblebees and deer, Nature Letters 449, 1044- 1049 (2007).
[54] G. M. Viswanathan, Sergey V. Buldyrev, Shlomo Havlin, M. G. E. da Luz, E. P. Ra- posok and H. Eugene Stanley, Optimizing the success of random searches, Nature 401, 911-914 (1999).
[55] Frederic Bartumeus, Francesc Peters, Salvador Pueyo, Cèlia Marrase and Jordi Cata- lan, Helical Lévy walks: Adjusting searching statistics to resource availability in microzoo- plankton, PNAS 100, 12771-12775 (2003).
[56] Sandra Cristina Costa, Condutividade e Movimento de Carga espacial em Materiais Desor- denados, São Carlos - São Paulo (2000), Tese (Doutorado), Instituto de Física de São Carlos - Departamento de Física e Ciência dos Materiais - Universidade de São Paulo. [57] [http://pt.wikipedia.org/wiki/Memória].
[58] G. M. Schültz and S. Trimper, Elephants can always remember: Exact long-range memory effects in a non-Markovian random walkPhys. Rev. E 70, 045101 (2004).
[59] M. A. A. da Silva, J. C. Cressoni, and G. M. Viswanathan, Discrete-time non-Markovian random walks: The effect of memory limitations on scaling, Physica A 364, 70 (2006).
[60] J.C. Cressoni, M.A.A. da Silva and G.M. Viswanathan, Amnestically Induced Persistence in Random Walks, Phys. Rev. Lett. 98, 070603 (2007).
[61] G. M. Borges, A. S. Ferreira, M. A. A. Silva, J. C. Cressoni, G. M. Viswanathan and A. M. Mariz, Superdiffusion in a non-Markovian random walk model with a Gaussian memory profile, Eur. Phys. J. B 85, 310 (2012).
[62] Marco Antonio Alves da Silva, G. M. Viswanathan, A. S. Ferreira and J. C. Cres- soni, Spontaneous symmetry breaking in amnestically induced persistence, Phys. Rev. E 77, 040101 (2008).
[63] K. Pearson, The problem of the random walk, Nature 72, 294 (1905).
[64] H. Scher and M. Lax, Stochastic Transport in a Disordered Solid. I. Theory, Phys. Rev. B 7, 4491-4502 (1973).
[65] H. Scher and E. Montroll, Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B 12, 2455-2477 (1975).
[66] E. W. Montroll and M. F. Shlesinger, The Wonderful World of Random Walks in None- quilibrium Phenomena II: From Stochastics to Hydrodynamics, eds. J. L. Lebowitz and E. W. Montroll North Holland, Amsterdam, vol. XI of Studies in Statistical Mechanics, p 1-121 (1984).
[67] G. Weiss, Aspects and Applications of the Random Walk, North Holland (Amsterdam New York Oxford), (1994).
[68] M. F. Shlesinger, B. J. West and J. Klafter, Lévy dynamics of enhanced diffusion: Applica- tion to e turbulence, Phys. Rev. Lett. 58, 1100 (1987).
[69] J. Masoliver, M. Montero and G. H. Weiss, Continuous-time random-walk model for fi- nancial distributions, Phys. Rev. E 67 021112 (2003).
[70] E. Ising, Beitrag zur Theorie des Ferromagnetismus, Z. Phys. 31, 253 (1925).
[71] V. M. Kenkre, Analytic Formulation, Exact Solutions, and Generalizations of the Elephant and the Alzheimer Random Walks, 6 Aug 2007, arXiv:0708.0034v2 [cond-mat.stat-mech].
[72] M. L. Felisberto, F. S. Passos, A. S. Ferreira, M. A. A. Silva, J. C. Cressoni and G. M. Viswanathan, Sudden onset of log-periodicity and superdiffusion in non-Markovian ran- dom walks with amnestically induced persistence: exact results, Eur. Phys. J. B 72 427-433 (2009).
[73] M. A. A. Silva, J. C. Cressoni, G. M. Schültz, G. M. Viswanathan and S. Trimper, Non- Gaussian propagator for elephant random walks, Phys. Rev. E 88, 022115 (2013).
[74] M. A. A. da Silva, G. M. Viswanathan, A. S. Ferreira and J. C. Cressoni, Spontaneous symmetry breaking in amnestically induced persistence, Phys. Rev. E 77, 040101R (2008). [75] Arlan da Silva Ferreira, Expoente de Hurst e Diagrama de Fase para Persistência Induzida
Amnesticamente em Processos não-Markovianos, Maceió-AL, Tese (Doutorado), Instituto de Física - Universidade Federal de Alagoas, (2009).
[76] Benoit B. Mandelbrot and John W. Van Ness, Fractional Brownian Motions, Fractional Noises and Applications, SIAM Rev. 10, 422-437 (1968).
[77] V. Tejedor and R. Metzler, Anomalous diffusion in correlated continuous time random walks, J. Phys. A: Math. Theor. 43, 082002 (2010).
[78] M. Magdziarz, R. Metzler, W. Szczotka and P. Zebrowski, Correlated continuous-time random walksscaling limits and Langevin picture, J. Stat. Mech., P04010 (2012).
[79] Constantino Tsallis, Renio S. Mendes, A.R. Plastinoa, The role of constraints within generalized nonextensive statistics, Physica A 261, 534?554 (1998).
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J
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Stat.
Mech.
(2014)
P04026
Superdiffusion driven by exponentially
decaying memory
G A Alves
1,2, J M de Ara´ujo
1, J C Cressoni
3,4,
L R da Silva
1, M A A da Silva
3and G M Viswanathan
11 Departamento de F´ısica Te´orica e Experimental, Universidade Federal do
Rio Grande do Norte, 59078-900, Natal, Rio Grande do Norte, Brazil
2 Departamento de F´ısica, Universidade Estadual do Piau´ı, 64002-150,
Teresina, Piau´ı, Brazil
3 Departamento de F´ısica e Qu´ımica, FCFRP, Universidade de S˜ao Paulo,
14040-903, Riber˜ao Preto, S˜ao Paulo, Brazil
4 Instituto de F´ısica, Universidade Federal de Alagoas, 57072-970, Macei´o,
Alagoas, Brazil
E-mail: [email protected],[email protected],
[email protected],[email protected],[email protected] and
Received 29 January 2014
Accepted for publication 6 March 2014 Published 25 April 2014
Online at stacks.iop.org/JSTAT/2014/P04026 doi:10.1088/1742-5468/2014/04/P04026
Abstract. A superdiffusive random walk model with exponentially decaying memory is reported. This seems to be a self-contradictory statement, since it is well known that random walks with exponentially decaying temporal correlations can be approximated arbitrarily well by Markov processes and that central limit theorems prohibit superdiffusion for Markovian walks with finite variance of step sizes. The solution to the apparent paradox is that the model is genuinely non-Markovian, due to a time-dependent decay constant associated with the exponential behavior.
Keywords: phase transformations (theory), stochastic processes (theory), diffusion
c
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Contents 1. Introduction 2 2. Model 3 3. Results 44. Discussion and conclusion 7
Acknowledgments 9
References 9
1. Introduction
Diffusion processes and random walks have been extensively used to describe important phenomena in many areas, such as physics, chemistry and biology [1]. The random walk and its generalization, the continuous time random walk model introduced by Montroll and Weiss in 1965 [2], are important tools for the study of many physical phenomena, such as in disordered media [3]–[6], earthquake modeling [7] and financial markets [8].
A basic fact which physicists learn early in their careers is that exponentially decaying correlations cannot lead to long-range order. For example, the Ising model with nearest neighbor interactions in one dimension cannot sustain long-range order at nonzero temperatures [9]. For the same reason, random walks with exponentially decaying correlations and whose step sizes have finite variance behave similarly to standard uncorrelated Brownian random walks at long times, with the mean squared displacement scaling linearly with time. Moreover, any random walk model with exponentially decaying memory can be modeled as an n-step Markov process, i.e. a Markov process in which the current transition probability depends only on the previous n steps taken. No matter how large we choose n, at long times the mean squared displacement necessarily scales linearly in time because of the central limit theorem. Specifically, at large times the memory becomes negligible so that upon renormalizing, i.e. coarse-graining, one recovers the uncorrelated Brownian random walk as a fixed point attractor of the renormalization flow map, so anomalous diffusion [10]–[19] is not possible. These are well known facts. It was thus a surprise to us when we found an apparent counter-example. We report here a random walk model with exponentially decaying memory which is superdiffusive even at long times, i.e. the mean squared displacement grows superlinearly in time, rather than linearly.
The resolution of the (apparent) paradox reveals a gap in how the subject is usually considered. Indeed, we show by construction that it is in fact possible to have a genuinely non-Markovian random walk model with exponentially decaying memory, provided that the decay constant is time-dependent.
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The model we study is a variant of the so-called elephant random walk (ERW) model proposed by Sch¨utz and Trimper [20]. The random walker keeps a record of the entire history of the walk, so that the walk is non-Markovian in principle. Many variants of this model have been proposed, such as the ‘Alzheimer walk’ model which led to the unexpected findings of amnestically induced superdiffusion and log-periodic superdiffusion (e.g., see [14]). Here, we propose a model with an exponentially decaying memory profile. This model is inspired by another recently proposed model [21] which had a (truncated) Gaussian memory profile. In this work we essentially replace the Gaussian by an exponential.
The ERW model, using the notation introduced in [20], starts at the origin at time t0 = 0 and retains memory of its complete history. In each time step the walker moves
one step to either the right or the left, i.e.,
xt+1 = xt+ vt+1 (1)
where vt+1 represents a stochastic noise with two-point autocorrelations (i.e. memory).
The walker can remember the entire history of prior random walk step directions {vt′} for t′
≤ t. At time t, one randomly chooses a random time 1 ≤ t′
≤ t with equal a priori probabilities. The current step direction vt is then chosen based on the value of vt′ as
vt+1=+vt
′, with probability p −vt′, with probability 1 − p.
(2) Without loss of generality, it is assumed that the first step always goes to the right, i.e. v1 = +1. The position at time t thus follows
xt = t
X
t′=1
vt′ (3)
and the second moment is given by
hx2ti = t/ (3 − 4p) , p < 3/4 t ln t, p = 3/4 t4p−2/ [(4p − 3)Γ(4p − 2)] , p > 3/4 (4)
which are exact relations valid in the asymptotic limit. The ERW presents a superdiffusive regime (p > 3/4) and a localized regime (p < 3/4), with p = 3/4 being marginally superdiffusive. Interestingly, for 1/2 < p < 3/4, the square of the mean does diverge, but more slowly than the mean square displacement, so that the behavior remains diffusive. This regime is termed an escape regime with a mean displacement given by hxti ∼ t2p−1.
The exact propagator is reported to be a Gaussian distribution [20], i.e., P (x, t) = 1 p4πD(t)exp −(x − hx(t)i) 2 4tD(t) (5) where D(t, p) = (1/8p−6)[(t/t0)4p−3−1] is the time- and p-dependent diffusive coefficient.
Within the superdiffusive regime the distribution has been found to be non-Gaussian [24].
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but with more recent events remembered more frequently—or easily—than those from the more distant past. We shall refer to this model as the exponential memory model. While in the ERW model the previous time t′
is chosen from a uniform distribution, in the exponential memory model t′
is randomly chosen from an exponential probability distribution. The probability of choosing a previous time t′
is then given by Pλ(t ′ , t) = A exp −λ(t − t′ ) t , (6)
where A is a normalization constant. The parameter λ adjusts the shape of the exponential distribution in the usual manner, but unlike typical decay constants λ is adimensional.
Unfortunately, this model does not yet have a known exact solution. Nevertheless, an approximate solution can be found by assuming that the exponential memory pattern can be mapped onto an equivalent memory profile with a—smaller—rectangular window size. Within our approach, this basic ansatz is necessary to get an approximate exact solution to the non-Markovian exponential memory model. Its validity is supported by the numerical results shown below. A non-Markovian rectangular memory profile model has a fixed memory size L = f t, where 0 < f < 1 is a new parameter that fixes the size of the memory. The probability of choosing a previous t′
is given simply by 1/L for (1 − f )t < t′
≤ t, and zero otherwise. This memory profile is flat, or constant, and has the shape of a rectangle instead of an exponential, such that the more ancient memories, i.e., those that occurred prior to time (1 − f )t, are forgotten. The random walker can therefore recall only a fraction f of the more recent steps.
3. Results
The main idea now is to determine an effective fraction feff(λ) which makes the model
with a rectangular memory pattern with f = feff behave just the same as the exponential
memory model with a given λ. Then the Fokker–Planck equation should be equivalent for both models, following the ideas discussed in [23]. We can define the memory’s effective length for the exponential memory model by
L ≡ Z t 0 [Pλ(t ′ , t)/Pmax(t ′ , t)] dt′ (7) where Pmax(t′, t) is the maximum value of Pλ(t′, t). Using (6) we get
L = Z t 0 e−λ(t−t′)/t dt′ = 1 − e−λ λ t (8)
which gives, using L = fefft,
feff = (1 − e −λ
)/λ. (9)
Equation (9) is the key result that allows us to perform the mapping between the two models and achieve an approximate solution for the exponential memory model by reducing it to a rectangular memory model. We shall refer to the model with a rectangular
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before [22], and its solution can provide a good analytical solution for the exponential memory model. The analytical solution can be obtained by first defining nf(t) and nb(t)
as the numbers of steps taken forward and backward, respectively, up to a given time t. Therefore, the total number of steps taken forward within the time interval [t − L, t] can be written as ∆nf = nf(t) − nf(t − L). Similarly, the total number of steps taken
backward in the same time interval is written as ∆nb = nb(t) − nb(t − L). Thus, the
effective probabilities of taking a step forward and backward, i.e., Peff+(t, x) and P− eff(t, x),
respectively, for t > 0 are given by
Peff+(t, x) = (∆nf/L)p + (∆nb/L)(1 − p) (10)
P−
eff(t, x) = (∆nb/L)p + (∆nf/L)(1 − p). (11)
Taking the difference between equations (10) and (11), with nf(t) + nb(t) = t and
nf(t − L) + nb(t − L) = t − L, we obtain an expression for the effective, or expected,
value of v at time t + 1, i.e., veff
t+1= Peff+(t, x) − P −
eff(t, x). (12)
Therefore, we can write ∆nf + ∆nb = L and xt = nf(t) − nb(t) + x0, and also x(t−L) =
nf(t − L) − nb(t − L) + x0, which gives xt− x(t−L) = ∆nf − ∆nb. Then we have ∆nf =
[L + xt− x(t−L)]/2 and ∆nb = [L − (xt− x(t−L))]/2. We can thus rewrite equation (12) as
vefft+1= αxt− x(t−L)
L (13)
where α = 2p − 1.
The conditional probability that the walker is at the position x at time t + 1 given the earlier position x0 at t = 0 is given by
P (x, t + 1|x0, 0) = P (x + 1, t|x0, 0)P −
(t, x + 1) + P (x − 1, t|x0, 0)P+(t, x − 1). (14)
Now using ∆nf+ ∆nb= L and ∆nf− ∆nb= xt− xt−L = x − G(x) and also the definitions
(10) and (11) again, we obtain Peff+(t, x) = 1 2 1 + α(x − G(x)) L (15) P− eff(t, x) = 1 2 1 − α(x − G(x)) L . (16)
Substitution of equations (15) and (16) into (14) gives P (x, t + 1|x0, 0) = 1 2 1 − α(x + 1 − G(x + 1)) L P (x + 1, t|x0, 0) +1 2 1 + α(x − 1 − G(x − 1)) L P (x − 1, t|x0, 0). doi:10.1088/1742-5468/2014/04/P04026 5
J
.
Stat.
Mech.
(2014)
P04026
subtracting P (x, t) from both sides of the expression above we obtain P (x, t + 1) − P (x, t) = P (x + 1, t) − 2P (x, t) + P (x − 1, t) 2 −α L [x + 1 − G(x + 1)] P (x + 1, t) − [x − 1 − G(x − 1)] P (x − 1, t) 2 . For large t we can write L = f t and G(x) = x(1−f )t. Thus, in the continuum limit, taken
in the usual manner, we can obtain an approximated FP equation for the propagator [24], i.e., ∂P (x, t) ∂t = 1 2 ∂2P (x, t) ∂x2 − α f t ∂ ∂xxP (x, t) − x(1−f )tP (x, t) . (17) The displacement x(1−f )t in equation (17) can be correlated with the displacement x = xt
by writing x(1−f )t = xh1−f(x, t), which defines the stochastic function h1−f(x, t). Notice
that this function can assume non-positive values. Using this definition, we can write the