Brownian motion 18.S995 - L03 & 04
Typical length scales http://www2.estrellamountain.edu/faculty/farabee/BIOBK/biobookcell2.html dunkel@math.mit.edu
Brownian motion
“Brownian” motion Jan Ingen-Housz (1730-1799) 1784/1785: http://www.physik.uni-augsburg.de/theo1/hanggi/History/BM-History.html
Robert Brown (1773-1858) Linnean society, London irregular motion of pollen in fluid 1827: irregul¨ are Eigenbewegung von Pollen in Fl¨ ussigkeit http://www.brianjford.com/wbbrownc.htm
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Nobel prize Jean Baptiste Perrin (1870-1942, Nobelpreis 1926) � colloidal particles of radius 0.53 µ m � successive positions every 30 seconds joined by straight line segments eculaire , Annales de chimie et de Mouvement brownien et r´ ealit´ e mol´ � mesh size is 3.2 µ m physique VIII 18, 5-114 (1909) Les Atomes , Paris, Alcan (1913) experimental evidence for Experimenteller Nachweis der atomistischen Struktur der Materie atomistic structure of matter
Norbert Wiener (1894-1864) MIT
Relevance in biology • intracellular transport • intercellular transport • microorganisms must beat BM to achieve directed locomotion • tracer diffusion = important experimental “tool” • generalized BMs (polymers, membranes, etc.) dunkel@math.mit.edu
Polymers & filaments (D=1) Dogic Lab, Brandeis Physical parameters (e.g. bending rigidity) from fluctuation analysis Drosophila oocyte Goldstein lab, PNAS 2012 dunkel@math.mit.edu
Brownian tracer particles in a bacterial suspension Bacillus subtilis Tracer colloids PRL 2013
http://web.mit.edu/mbuehler/www/research/f103.jpg
Basic idea Split dynamics into • deterministic part (drift) • random part (diffusion)
Typical problems http://www.pnas.org/content/104/41/16098/F1.expansion.html Determine D • noise ‘structure’ • transport coefficients • first passage (escape) times
Probability space [0 , 1] A B P [ ∅ ] = 0
Expectation values of discrete random variables
Expectation values of continuous random variables
Random walk model N X X N = x 0 + ` S i i =1
1.1 Random walks 1.1.1 Unbiased random walk (RW) Consider the one-dimensional unbiased RW (fixed initial position X 0 = x 0 , N steps of length ` ) N X X N = x 0 + ` S i (1.1) i =1 where S i ∈ {± 1 } are iid. random variables (RVs) with P [ S i = ± 1] = 1 / 2. Noting that 1 − 1 · 1 2 + 1 · 1 E [ S i ] = 2 = 0 , (1.2) ( − 1) 2 · 1 2 + (1) 2 · 1 � � ij E [ S 2 E [ S i S j ] = i ] = � ij = � ij , (1.3) 2 we find for the first moment of the RW N X E [ X N ] = x 0 + ` E [ S i ] = x 0 (1.4) i =1
1.1 Random walks 1.1.1 Unbiased random walk (RW) Consider the one-dimensional unbiased RW (fixed initial position X 0 = x 0 , N steps of length ` ) N X X N = x 0 + ` S i (1.1) i =1 where S i ∈ {± 1 } are iid. random variables (RVs) with P [ S i = ± 1] = 1 / 2. Noting that 1 − 1 · 1 2 + 1 · 1 E [ S i ] = 2 = 0 , (1.2) ( − 1) 2 · 1 2 + (1) 2 · 1 � � ij E [ S 2 E [ S i S j ] = i ] = � ij = � ij , (1.3) 2 we find for the first moment of the RW N X E [ X N ] = x 0 + ` E [ S i ] = x 0 (1.4) i =1
1.1 Random walks 1.1.1 Unbiased random walk (RW) Consider the one-dimensional unbiased RW (fixed initial position X 0 = x 0 , N steps of length ` ) N X X N = x 0 + ` S i (1.1) i =1 where S i ∈ {± 1 } are iid. random variables (RVs) with P [ S i = ± 1] = 1 / 2. Noting that 1 − 1 · 1 2 + 1 · 1 E [ S i ] = 2 = 0 , (1.2) ( − 1) 2 · 1 2 + (1) 2 · 1 � � ij E [ S 2 E [ S i S j ] = i ] = � ij = � ij , (1.3) 2 we find for the first moment of the RW N X E [ X N ] = x 0 + ` E [ S i ] = x 0 (1.4) i =1
Second moment (uncentered) N X E [ X 2 S i ) 2 ] N ] = E [( x 0 + ` i =1 N N N X X X E [ x 2 S i + ` 2 = 0 + 2 x 0 ` S i S j ] i =1 i =1 j =1 N N X X x 2 0 + 2 x 0 · 0 + ` 2 = E [ S i S j ] i =1 j =1 N N X X x 2 0 + 2 x 0 · 0 + ` 2 = � ij i =1 j =1 x 2 0 + ` 2 N. = (1.5)
Second moment (uncentered) N X E [ X 2 S i ) 2 ] N ] = E [( x 0 + ` i =1 N N N X X X E [ x 2 S i + ` 2 = 0 + 2 x 0 ` S i S j ] i =1 i =1 j =1 N N X X x 2 0 + 2 x 0 · 0 + ` 2 = E [ S i S j ] i =1 j =1 N N X X x 2 0 + 2 x 0 · 0 + ` 2 = � ij i =1 j =1 x 2 0 + ` 2 N. = (1.5)
Second moment (uncentered) N X E [ X 2 S i ) 2 ] N ] = E [( x 0 + ` i =1 N N N X X X E [ x 2 S i + ` 2 = 0 + 2 x 0 ` S i S j ] i =1 i =1 j =1 N N X X x 2 0 + 2 x 0 · 0 + ` 2 = E [ S i S j ] i =1 j =1 N N X X x 2 0 + 2 x 0 · 0 + ` 2 = � ij i =1 j =1 x 2 0 + ` 2 N. = (1.5)
Second moment (uncentered) N X E [ X 2 S i ) 2 ] N ] = E [( x 0 + ` i =1 N N N X X X E [ x 2 S i + ` 2 = 0 + 2 x 0 ` S i S j ] i =1 i =1 j =1 N N X X x 2 0 + 2 x 0 · 0 + ` 2 = E [ S i S j ] i =1 j =1 N N X X x 2 0 + 2 x 0 · 0 + ` 2 = � ij i =1 j =1 x 2 0 + ` 2 N. = (1.5)
Second moment (uncentered) N X E [ X 2 S i ) 2 ] N ] = E [( x 0 + ` i =1 N N N X X X E [ x 2 S i + ` 2 = 0 + 2 x 0 ` S i S j ] i =1 i =1 j =1 N N X X x 2 0 + 2 x 0 · 0 + ` 2 = E [ S i S j ] i =1 j =1 N N X X x 2 0 + 2 x 0 · 0 + ` 2 = � ij i =1 j =1 x 2 0 + ` 2 N. = (1.5)
Continuum limit N X X N = x 0 + ` S i i =1 Let
Continuum limit ◆ N ✓ N ✓ 1 ◆ P ( N, K ) = N − K 2 2 ◆ N ✓ 1 N ! = (( N + K ) / 2)! (( N − K ) / 2)! . (1.8) 2 The associated probability density function (PDF) can be found by defining p ( t, x ) := P ( N, K ) = P ( t/ ⌧ , x/ ` ) (1.9) 2 ` 2 ` and considering limit ⌧ , ` → 0 such that D := ` 2 2 ⌧ = const, (1.10)
Continuum limit (pset1) yielding the Gaussian r � x 2 1 ✓ ◆ p ( t, x ) ' 4 π Dt exp (1.11) 4 Dt Eq. (1.11) is the fundamental solution to the di ff usion equation, ∂ t p t = D ∂ xx p, (1.12) where ∂ t , ∂ x , ∂ xx , . . . denote partial derivatives. The mean square displacement of the con- tinuous process described by Eq. (1.11) is Z dx x 2 p ( t, x ) = 2 Dt, E [ X ( t ) 2 ] = (1.13) in agreement with Eq. (1.7).
One often classifies di ff usion processes by the (asymptotic) power-law growth Remark of the mean square displacement, E [( X ( t ) � X (0)) 2 ] ⇠ t µ . (1.14) • µ = 0 : Static process with no movement. • 0 < µ < 1 : Sub-di ff usion, arises typically when waiting times between subsequent jumps can be long and/or in the presence of a su ffi ciently large number of obstacles (e.g. slow di ff usion of molecules in crowded cells). • µ = 1 : Normal di ff usion, corresponds to the regime governed by the standard Central Limit Theorem (CLT). • 1 < µ < 2 : Super-di ff usion, occurs when step-lengths are drawn from distributions with infinite variance (L´ evy walks; considered as models of bird or insect movements). • µ = 2 : Ballistic propagation (deterministic wave-like process).
non-Brownian Brownian motion Levy-flight
1.1.2 Biased random walk (BRW) Consider a one-dimensional hopping process on a discrete lattice (spacing ` ), defined such that during a time-step ⌧ a particle at position X ( t ) = ` j 2 ` Z can either (i) jump a fixed distance ` to the left with probability � , or (ii) jump a fixed distance ` to the right with probability ⇢ , or (iii) remain at its position x with probability (1 � � � ⇢ ). Assuming that the process is Markovian (does not depend on the past), the evolution of the associated probability vector P ( t ) = ( P ( t, x )) = ( P j ( t )), where x = ` j , is governed by the master equation P ( t + ⌧ , x ) = (1 � � � ⇢ ) P ( t, x ) + ⇢ P ( t, x � ` ) + � P ( t, x + ` ) . (1.15)
Master equations P ( t + ⌧ , x ) = (1 � � � ⇢ ) P ( t, x ) + ⇢ P ( t, x � ` ) + � P ( t, x + ` ) . (1.15) Technically, ⇢ , � and (1 � � � ⇢ ) are the non-zero-elements of the corresponding transition matrix W = ( W ij ) with W ij > 0 that governs the evolution of the column probability vector P ( t ) = ( P j ( t )) = ( P ( t, y )) by P i ( t + ⌧ ) = W ij P j ( t ) (1.16a) or, more generally, for n steps P ( t + n ⌧ ) = W n P ( t ) . (1.16b) The stationary solutions are the eigenvectors of W with eigenvalue 1. To preserve normal- ization, one requires P i W ij = 1.
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