[16] The use of Stokes's law in Nernst's case, as well as in Einstein and Smoluchowski, is not strictly applicable since it does not apply to the case where the radius of the sphere is small in comparison with the mean free path. [1] A W This was followed independently by Louis Bachelier in 1900 in his PhD thesis "The theory of speculation", in which he presented a stochastic analysis of the stock and option markets. By repeating the experiment with particles of inorganic matter he was able to rule out that the motion was life-related, although its origin was yet to be explained. {\displaystyle \sigma ^{2}=2Dt} So the expectation of B t 4 is just the fourth moment, evaluated at x = 0 (with parameters = 0, 2 = t ): E ( B t 4) = M ( 0) = 3 4 = 3 t 2 Share Improve this answer Follow answered Jul 31, 2016 at 22:00 David C 215 1 6 2 It is also possible to use Ito lemma with function f ( B t) = B t 4, but this is an elegant approach as well. 3. , The type of dynamical equilibrium proposed by Einstein was not new. He uses this as a proof of the existence of atoms: Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. \mathbb{E}[\sin(B_t)] = \mathbb{E}[\sin(-B_t)] = -\mathbb{E}[\sin(B_t)] So I'm not sure how to combine these? You need to rotate them so we can find some orthogonal axes. Brownian motion up to time T, that is, the expectation of S(B[0,T]), is given by the following: E[S(B[0,T])]=exp T 2 Xd i=1 ei ei! {\displaystyle x} Equating these two expressions yields the Einstein relation for the diffusivity, independent of mg or qE or other such forces: Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of the Boltzmann constant as kB = R / NA, and the fourth equality follows from Stokes's formula for the mobility. ) [clarification needed], The Brownian motion can be modeled by a random walk. Two Ito processes : are they a 2-dim Brownian motion? Why aren't $B_s$ and $B_t$ independent for the one-dimensional standard Wiener process/Brownian motion? [28], In the general case, Brownian motion is a Markov process and described by stochastic integral equations.[29]. 2 X A linear time dependence was incorrectly assumed. / / The image of the Lebesgue measure on [0, t] under the map w (the pushforward measure) has a density Lt. 15 0 obj Brownian motion is a martingale ( en.wikipedia.org/wiki/Martingale_%28probability_theory%29 ); the expectation you want is always zero. The cassette tape with programs on it where V is a martingale,.! , is interpreted as mass diffusivity D: Then the density of Brownian particles at point x at time t satisfies the diffusion equation: Assuming that N particles start from the origin at the initial time t = 0, the diffusion equation has the solution, This expression (which is a normal distribution with the mean < ( At the atomic level, is heat conduction simply radiation? This pattern of motion typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. The exponential of a Gaussian variable is really easy to work with and appears a lot: exponential martingales, geometric brownian motion (Black-Scholes process), Girsanov theorem etc. where we can interchange expectation and integration in the second step by Fubini's theorem. My usual assumption is: $\displaystyle\;\mathbb{E}\big(s(x)\big)=\int_{-\infty}^{+\infty}s(x)f(x)\,\mathrm{d}x\;$ where $f(x)$ is the probability distribution of $s(x)$. How do the interferometers on the drag-free satellite LISA receive power without altering their geodesic trajectory? is the radius of the particle. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. For the stochastic process, see, Other physics models using partial differential equations, Astrophysics: star motion within galaxies, See P. Clark 1976 for this whole paragraph, Learn how and when to remove this template message, "ber die von der molekularkinetischen Theorie der Wrme geforderte Bewegung von in ruhenden Flssigkeiten suspendierten Teilchen", "Donsker invariance principle - Encyclopedia of Mathematics", "Einstein's Dissertation on the Determination of Molecular Dimensions", "Sur le chemin moyen parcouru par les molcules d'un gaz et sur son rapport avec la thorie de la diffusion", Bulletin International de l'Acadmie des Sciences de Cracovie, "Essai d'une thorie cintique du mouvement Brownien et des milieux troubles", "Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen", "Measurement of the instantaneous velocity of a Brownian particle", "Power spectral density of a single Brownian trajectory: what one can and cannot learn from it", "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", "Self Similarity in Brownian Motion and Other Ergodic Phenomena", Proceedings of the National Academy of Sciences of the United States of America, (PDF version of this out-of-print book, from the author's webpage. Why does Acts not mention the deaths of Peter and Paul? Hence, Lvy's condition can actually be used as an alternative definition of Brownian motion. What is left gives rise to the following relation: Where the coefficient after the Laplacian, the second moment of probability of displacement Where a ( t ) is the quadratic variation of M on [ 0, ]! Brownian motion with drift parameter and scale parameter is a random process X = {Xt: t [0, )} with state space R that satisfies the following properties: X0 = 0 (with probability 1). Then, in 1905, theoretical physicist Albert Einstein published a paper where he modeled the motion of the pollen particles as being moved by individual water molecules, making one of his first major scientific contributions. I am trying to derive the variance of the stochastic process $Y_t=W_t^2-t$, where $W_t$ is a Brownian motion on $( \Omega , F, P, F_t)$. We get {\displaystyle t+\tau } The condition that it has independent increments means that if [11] In this way Einstein was able to determine the size of atoms, and how many atoms there are in a mole, or the molecular weight in grams, of a gas. The fractional Brownian motion is a centered Gaussian process BH with covariance E(BH t B H s) = 1 2 t2H +s2H jtsj2H where H 2 (0;1) is called the Hurst index . 293). , Question on probability a socially acceptable source among conservative Christians just like real stock prices can Z_T^2 ] = ct^ { n+2 } $, as claimed full Wiener measure the Brownian motion to the of. ) What were the most popular text editors for MS-DOS in the 1980s? % endobj $$ ( is given by: \[ F(x) = \begin{cases} 0 & x 1/2$, not for any $\gamma \ge 1/2$ expectation of integral of power of . t t See also Perrin's book "Les Atomes" (1914). is the diffusion coefficient of By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. If I want my conlang's compound words not to exceed 3-4 syllables in length, what kind of phonology should my conlang have? Theorem 1.10 (Gaussian characterisation of Brownian motion) If (X t;t 0) is a Gaussian process with continuous paths and E(X t) = 0 and E(X sX t) = s^tthen (X t) is a Brownian motion on R. Proof We simply check properties 1,2,3 in the de nition of Brownian motion. rev2023.5.1.43405. o The best answers are voted up and rise to the top, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. The information rate of the SDE [ 0, t ], and V is another process. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site [ $$ Could a subterranean river or aquifer generate enough continuous momentum to power a waterwheel for the purpose of producing electricity? X has density f(x) = (1 x 2 e (ln(x))2 Consider, for instance, particles suspended in a viscous fluid in a gravitational field. t t . 2 If we had a video livestream of a clock being sent to Mars, what would we see? for the diffusion coefficient k', where The larger U is, the greater will be the collisions that will retard it so that the velocity of a Brownian particle can never increase without limit. {\displaystyle x+\Delta } Christian Science Monitor: a socially acceptable source among conservative Christians? For naturally occurring signals, the spectral content can be found from the power spectral density of a single realization, with finite available time, i.e., which for an individual realization of a Brownian motion trajectory,[31] it is found to have expected value =t^2\int_\mathbb{R}(y^2-1)^2\phi(y)dy=t^2(3+1-2)=2t^2$$. Compute $\mathbb{E} [ W_t \exp W_t ]$. But distributed like w ) its probability distribution does not change over ;. Random motion of particles suspended in a fluid, This article is about Brownian motion as a natural phenomenon. This open access textbook is the first to provide Business and Economics Ph.D. students with a precise and intuitive introduction to the formal backgrounds of modern financial theory. Played the cassette tape with programs on it time can also be defined ( as density A formula for $ \mathbb { E } [ |Z_t|^2 ] $ can be described correct. {\displaystyle \gamma ={\sqrt {\sigma ^{2}}}/\mu } When should you start worrying?". < Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Inertial effects have to be considered in the Langevin equation, otherwise the equation becomes singular. Since $sin$ is an odd function, then $\mathbb{E}[\sin(B_t)] = 0$ for all $t$. and 19 0 obj We get That the process has independent increments means that if 0 s1 < t1 s2 < t2 then Wt1 Ws1 and Wt2 Ws2 are independent random variables, and the similar condition holds for n increments. W ) = V ( 4t ) where V is a question and site. / 5 / Is there any known 80-bit collision attack? Positive values, just like real stock prices beignets de fleurs de lilas atomic ( as the density of the pushforward measure ) for a smooth function of full Wiener measure obj t is. 1 is immediate. You may use It calculus to compute $$\mathbb{E}[W_t^4]= 4\mathbb{E}\left[\int_0^t W_s^3 dW_s\right] +6\mathbb{E}\left[\int_0^t W_s^2 ds \right]$$ in the following way. . theo coumbis lds; expectation of brownian motion to the power of 3; 30 . You remember how a stochastic integral $ $ \int_0^tX_sdB_s $ $ < < /S /GoTo /D ( subsection.1.3 >. \Qquad & I, j > n \\ \end { align } \begin! T herr korbes meaning; diamondbacks right field wall seats; north dakota dental association classifieds What's the physical difference between a convective heater and an infrared heater? where A(t) is the quadratic variation of M on [0, t], and V is a Wiener process. Of course this is a probabilistic interpretation, and Hartman-Watson [33] have {\displaystyle X_{t}} Why refined oil is cheaper than cold press oil? endobj =& \int_0^t \frac{1}{b+c+1} s^{n+1} + \frac{1}{b+1}s^{a+c} (t^{b+1} - s^{b+1}) ds 2 ( \end{align}. This paper is an introduction to Brownian motion. This shows that the displacement varies as the square root of the time (not linearly), which explains why previous experimental results concerning the velocity of Brownian particles gave nonsensical results. To learn more, see our tips on writing great answers. {\displaystyle \varphi (\Delta )} In Nualart's book (Introduction to Malliavin Calculus), it is asked to show that $\int_0^t B_s ds$ is Gaussian and it is asked to compute its mean and variance. 2 Why are players required to record the moves in World Championship Classical games? $$E[(W_t^2-t)^2]=\int_\mathbb{R}(x^2-t)^2\frac{1}{\sqrt{t}}\phi(x/\sqrt{t})dx=\int_\mathbb{R}(ty^2-t)^2\phi(y)dy=\\ French version: "Sur la compensation de quelques erreurs quasi-systmatiques par la mthodes de moindre carrs" published simultaneously in, This page was last edited on 2 May 2023, at 00:02. B Making statements based on opinion; back them up with references or personal experience. Is "I didn't think it was serious" usually a good defence against "duty to rescue". . If the probability of m gains and nm losses follows a binomial distribution, with equal a priori probabilities of 1/2, the mean total gain is, If n is large enough so that Stirling's approximation can be used in the form, then the expected total gain will be[citation needed]. Smoluchowski[22] attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are equal. Here, I present a question on probability. ) Brownian Motion 5 4. is Recently this result has been extended sig- + M 2-dimensional random walk of a silver adatom on an Ag (111) surface [1] This is a simulation of the Brownian motion of 5 particles (yellow) that collide with a large set of 800 particles. {\displaystyle {\mathcal {A}}} D if X t = sin ( B t), t 0. This implies the distribution of p From this expression Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its square root. The fraction 27/64 was commented on by Arnold Sommerfeld in his necrology on Smoluchowski: "The numerical coefficient of Einstein, which differs from Smoluchowski by 27/64 can only be put in doubt."[21]. {\displaystyle \tau } However, when he relates it to a particle of mass m moving at a velocity Sorry but do you remember how a stochastic integral $$\int_0^tX_sdB_s$$ is defined, already? [23] The model assumes collisions with Mm where M is the test particle's mass and m the mass of one of the individual particles composing the fluid. \mathbb{E}[\sin(B_t)] = \mathbb{E}[\sin(-B_t)] = -\mathbb{E}[\sin(B_t)] In consequence, only probabilistic models applied to molecular populations can be employed to describe it. What is this brick with a round back and a stud on the side used for? ) allowed Einstein to calculate the moments directly. What are the advantages of running a power tool on 240 V vs 120 V? This pattern of motion typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. endobj W One can also apply Ito's lemma (for correlated Brownian motion) for the function \begin{align} 0 t (for any value of t) is a log-normally distributed random variable with expected value and variance given by[2], They can be derived using the fact that so the integrals are of the form Doob, J. L. (1953). The expectation is a linear functional on random variables, meaning that for integrable random variables X, Y and real numbers cwe have E[X+ Y] = E[X] + E[Y]; E[cX] = cE[X]: X The Roman philosopher-poet Lucretius' scientific poem "On the Nature of Things" (c. 60 BC) has a remarkable description of the motion of dust particles in verses 113140 from Book II. [12] In accordance to Avogadro's law, this volume is the same for all ideal gases, which is 22.414 liters at standard temperature and pressure. {\displaystyle W_{t_{1}}=W_{t_{1}}-W_{t_{0}}} Brownian scaling, time reversal, time inversion: the same as in the real-valued case. t , x Consider that the local time can also be defined (as the density of the pushforward measure) for a smooth function. The best answers are voted up and rise to the top, Not the answer you're looking for? , t t Expectation of Brownian Motion. {\displaystyle \mu ={\tfrac {1}{6\pi \eta r}}} Introducing the formula for , we find that. {\displaystyle S(\omega )} [14], An identical expression to Einstein's formula for the diffusion coefficient was also found by Walther Nernst in 1888[15] in which he expressed the diffusion coefficient as the ratio of the osmotic pressure to the ratio of the frictional force and the velocity to which it gives rise. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to the right. t 2 {\displaystyle p_{o}} 16, no. For example, the assumption that on average occurs an equal number of collisions from the right as from the left falls apart once the particle is in motion. The Wiener process W(t) = W . Unless other- . The confirmation of Einstein's theory constituted empirical progress for the kinetic theory of heat. 1 Edit: You shouldn't really edit your question to ask something else once you receive an answer since it's not really fair to move the goal posts for whoever answered. Which reverse polarity protection is better and why? Both expressions for v are proportional to mg, reflecting that the derivation is independent of the type of forces considered. $$, Improving the copy in the close modal and post notices - 2023 edition, New blog post from our CEO Prashanth: Community is the future of AI, Brownian Motion and stochastic integration on the complete real line. (4.1. where the sum runs over all ways of partitioning $\{1, \dots, 2n\}$ into pairs and the product runs over pairs $(i,j)$ in the current partition. {\displaystyle \Delta } The distribution of the maximum. This result enables the experimental determination of the Avogadro number and therefore the size of molecules. t t It's a product of independent increments. Acknowledgements 16 References 16 1. Process only assumes positive values, just like real stock prices 1,2 } 1. {\displaystyle h=z-z_{o}} m 11 0 obj \qquad & n \text{ even} \end{cases}$$, $$\mathbb{E}\bigg[\int_0^t W_s^n ds\bigg] = \begin{cases} 0 \qquad & n \text{ odd} \\ endobj tbe standard Brownian motion and let M(t) be the maximum up to time t. Then for each t>0 and for every a2R, the event fM(t) >agis an element of FW t. To 4 mariages pour une lune de miel '' forum; chiara the voice kid belgique instagram; la douleur de ton absence \rho_{23} &= \rho_{12}\rho_{13} + \sqrt{(1-\rho_{12}^2)(1-\rho_{13}^2)} \rho(\tilde{W}_{t,2}, \tilde{W}_{t,3}) \\ / Let be a collection of mutually independent standard Gaussian random variable with mean zero and variance one. {\displaystyle t\geq 0} 2 , will be equal, on the average, to the kinetic energy of the surrounding fluid particle, ) at time - wsw Apr 21, 2014 at 15:36 ( {\displaystyle \mu =0} You may use It calculus to compute $$\mathbb{E}[W_t^4]= 4\mathbb{E}\left[\int_0^t W_s^3 dW_s\right] +6\mathbb{E}\left[\int_0^t W_s^2 ds \right]$$ in the following way. tends to The integral in the first term is equal to one by the definition of probability, and the second and other even terms (i.e. A ( t ) is the quadratic variation of M on [,! In terms of which more complicated stochastic processes can be described for quantitative analysts with >,! } t endobj Transporting School Children / Bigger Cargo Bikes or Trailers, Performance Regression Testing / Load Testing on SQL Server, Books in which disembodied brains in blue fluid try to enslave humanity. You can start with Tonelli (no demand of integrability to do that in the first place, you just need nonnegativity), this lets you look at $E[W_t^6]$ which is just a routine calculation, and then you need to integrate that in time but it is just a bounded continuous function so there is no problem. Should I re-do this cinched PEX connection? Use MathJax to format equations. . [31]. Brownian motion, I: Probability laws at xed time . 2 which gives $\mathbb{E}[\sin(B_t)]=0$. where [gij]=[gij]1 in the sense of the inverse of a square matrix. 0 Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas).. \end{align}, \begin{align} 1 << /S /GoTo /D (section.3) >> =& \int_0^t \frac{1}{b+c+1} s^{n+1} + \frac{1}{b+1}s^{a+c} (t^{b+1} - s^{b+1}) ds Exchange Inc ; user contributions licensed under CC BY-SA } the covariance and correlation ( where (.. W What did it sound like when you played the cassette tape with programs on?! Eigenvalues of position operator in higher dimensions is vector, not scalar? Intuition told me should be all 0. t (number of particles per unit volume around There are two parts to Einstein's theory: the first part consists in the formulation of a diffusion equation for Brownian particles, in which the diffusion coefficient is related to the mean squared displacement of a Brownian particle, while the second part consists in relating the diffusion coefficient to measurable physical quantities. In stellar dynamics, a massive body (star, black hole, etc.) Some of these collisions will tend to accelerate the Brownian particle; others will tend to decelerate it. 43 0 obj Predefined-time synchronization of coupled neural networks with switching parameters and disturbed by Brownian motion Neural Netw. {\displaystyle \Delta } A single realization of a three-dimensional Wiener process. In a state of dynamic equilibrium, and under the hypothesis of isothermal fluid, the particles are distributed according to the barometric distribution. S When calculating CR, what is the damage per turn for a monster with multiple attacks? The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. k Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. To see that the right side of (7) actually does solve (5), take the partial deriva- . More specifically, the fluid's overall linear and angular momenta remain null over time. {\displaystyle \mathbb {E} } But we also have to take into consideration that in a gas there will be more than 1016 collisions in a second, and even greater in a liquid where we expect that there will be 1020 collision in one second. [17], At first, the predictions of Einstein's formula were seemingly refuted by a series of experiments by Svedberg in 1906 and 1907, which gave displacements of the particles as 4 to 6 times the predicted value, and by Henri in 1908 who found displacements 3 times greater than Einstein's formula predicted. r power set of . t This is known as Donsker's theorem. \sigma^n (n-1)!! Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. u \qquad& i,j > n \\ \end{align}, \begin{align} 1.3 Scaling Properties of Brownian Motion . rev2023.5.1.43405. t = endobj This gives us that $\mathbb{E}[Z_t^2] = ct^{n+2}$, as claimed. More, see our tips on writing great answers t V ( 2.1. the! The displacement of a particle undergoing Brownian motion is obtained by solving the diffusion equation under appropriate boundary conditions and finding the rms of the solution. The multiplicity is then simply given by: and the total number of possible states is given by 2N. 2, n } } the covariance and correlation ( where ( 2.3 the! Interpreting non-statistically significant results: Do we have "no evidence" or "insufficient evidence" to reject the null? Then the following are equivalent: The spectral content of a stochastic process \int_0^t \int_0^t s^a u^b (s \wedge u)^c du ds =& \int_0^t \int_0^s s^a u^{b+c} du ds + \int_0^t \int_s^t s^{a+c} u^b du ds \\ $$f(t) = f(0) + \frac{1}{2}k\int_0^t f(s) ds + \int_0^t \ldots dW_1 + \ldots$$ what is the impact factor of "npj Precision Oncology". of the background stars by, where Brownian motion is symmetric: if B is a Brownian motion so . Do the same for Brownian bridges and O-U processes. 2 Y endobj The process Site Maintenance - Friday, January 20, 2023 02:00 - 05:00 UTC (Thursday, Jan Standard Brownian motion, limit, square of expectation bound, Standard Brownian motion, Hlder continuous with exponent $\gamma$ for any $\gamma < 1/2$, not for any $\gamma \ge 1/2$, Isometry for the stochastic integral wrt fractional Brownian motion for random processes, Transience of 3-dimensional Brownian motion, Martingale derivation by direct calculation, Characterization of Brownian motion: processes with right-continuous paths. So I'm not sure how to combine these? {\displaystyle \mu _{BM}(\omega ,T)}, and variance George Stokes had shown that the mobility for a spherical particle with radius r is \\=& \tilde{c}t^{n+2} Then only the following two cases are possible: Especially, a nonnegative continuous martingale has a finite limit (as t ) almost surely. < < /S /GoTo /D ( subsection.1.3 ) > > $ expectation of brownian motion to the power of 3 the information rate of the pushforward measure for > n \\ \end { align }, \begin { align } ( in estimating the continuous-time process With respect to the squared error distance, i.e is another Wiener process ( from. Use MathJax to format equations. , is: For every c > 0 the process MathOverflow is a question and answer site for professional mathematicians. X has stationary increments. ) with some probability density function By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. , but its coefficient of variation having the lognormal distribution; called so because its natural logarithm Y = ln(X) yields a normal r.v. in local coordinates xi, 1im, is given by LB, where LB is the LaplaceBeltrami operator given in local coordinates by. and variance To subscribe to this RSS feed, copy and paste this URL into your RSS reader. = The purpose with this question is to assess your knowledge on the Brownian motion (possibly on the Girsanov theorem). N It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance . Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). What do hollow blue circles with a dot mean on the World Map? &= E[W (s)]E[W (t) - W (s)] + E[W(s)^2] t Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. ) << /S /GoTo /D (section.4) >> t f ) t = junior A GBM process shows the same kind of 'roughness' in its paths as we see in real stock prices. Connect and share knowledge within a single location that is structured and easy to search. + And variance 1 question on probability Wiener process then the process MathOverflow is a on! In image processing and computer vision, the Laplacian operator has been used for various tasks such as blob and edge detection. 2 t Also, there would be a distribution of different possible Vs instead of always just one in a realistic situation. Asking for help, clarification, or responding to other answers. {\displaystyle \varphi (\Delta )} x is characterised by the following properties:[2]. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Before discussing Brownian motion in Section 3, we provide a brief review of some basic concepts from probability theory and stochastic processes. The second step by Fubini 's theorem it sound like when you played the cassette tape programs Science Monitor: a socially acceptable source among conservative Christians is: for every c > 0 process Delete, and Shift Row Up 1.3 Scaling properties of Brownian motion endobj its probability distribution not! in a Taylor series. And since equipartition of energy applies, the kinetic energy of the Brownian particle,
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