cosmicmolecularquanta does the inertia of a body depend on its energy content? annalen der physik,...
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Does the inertia of a body depend on its energy content? Annalen der Physik, 18(1905), pp. 639-41.
E = m c2
On the electrodynamics of moving bodies (special relativity) Annalen der Physik, 17(1905), pp. 891-921.
"On a heuristic viewpoint concerning the production and transformation of light." (light quantum/photoelectric effect paper) (17 March 1905) Annalen der Physik, 17(1905), pp. 132-148.
"On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat." (Brownian motion paper) (May 1905; received 11 May 1905) Annalen der Physik, 17(1905), pp. 549-560.
Brownian motion due to randomly directed impulses from collisions with thermally excited molecules
Random walks and diffusion
a
KTD
6
K = Boltzmann's constant 1.38 10-16 g cm2 s-2 oK-1
bacteriacolloids a ≈ 1 µ D ≈ 10-6 to 10-7 cm2/s
T = temperature in oK
µ = viscosity
Leads to diffusion
More importantly:
Shows how the kinematic* of small scale motion leads directly to macroscale properties
*Kinematics: how things move Dynamics: why things move
Random walk Diffusion
MolecularTurbulenceMotility of organisms
Random walks and diffusion
Microscopic Macroscopic
Property of "individual" Property of "population"
x1
x2
xn
X = x1 + x2 + ..... + xn
Each change in direction is totally random
xi = 0
X = 0
Ensemble averageFor 1 realization
The average position does not change
Random walks and diffusion: theory
Speed vRun duration
221 ... nxxxXX
....222
12
1
11
1
n
iii
n
iii
n
iii xxxxxxXX
Diffusion defined by
2
2
1
2
1
Ndt
d
ND XX
If is exponentially distributed then
22 2 2
2 11v
NND
Random walks and diffusion: theory
Uncorrelated= 0
tn
2
2 XXNumber of steps n =Total time t / run duration
Each path segment is of lenght v dt
Particle moves with speed v
Its position is updated every time step dt
The probability of changing direction in a time step is
dte dt /1
Random walks and diffusion: modelling
Random walk of a single particle
Random walk of a single particle
Random walks of a cluster of particles
)4/(2
4),( Dtxe
Dt
Ntx
2
2
1vD
Random walks of a cluster of particles: diffusion
Motility of organisms
erraticsinuoushelicalhop - sink
some element of randomness
Motility of organisms: flagellate Bodo designis
Motility of organisms: adult copepod temora longicornis
How to analyse these swimming paths ?
NGDR (net to gross distance ratio)
Fractal dimension ()
Diffusive analog of motility
Ciliate Balanion comatum
l
L
l L 1/
NGDR = l / L
l (2Dt)1/ (fractal dimension 2)
Diffusion by continuous motion
GI Taylor
Diffusion by continuous motion
1
x1
xn
x2
x3
Xn
a measure of how fast the path becomes de-correlated with itself
Diffusion by continuous motion
G.I. Taylor (1921); Diffusion by continuous random (Brownian) motion
l2 = 2 (L – (1 – e–L/))
Diffusion by continuous motion
l2 = 2 v2 (t – (1 – e–t/))
or equivalently
Diffusion by continuous motion
lL
l
L
log(t)
log(
l)
slope = 1
slope = 1/2
l = v t for t << Ballistic
l = (2 v t)1/2 for t >> Diffusive
Diffusion by continuous motion
Long termstochasticity
Diffusion by continuous motion
Short termcoherency
x, m
200 400 600 800 1000 1200
y,
m
0
200
400
600
800
1000
time (s)
0 2 4 6
net d
ispl
acem
ent ( m
)
0
100
200
300
400
500Ciliate Balanion comatum
Jakobsen et al (submitted)Visser & Kiørboe (submitted)
v = 220 ± 10 m/s = 0.3 ± 0.03 s = 65 ± 10 mD = 50 ± 10 x 10-6 cm2/s
Example
similar motility for organisms from bacteria to copepods
Diffusion by continuous motion
Motility length scale
Swimming speed v
Capture radius R
Ambush predator feeding on motile prey
simple example
Two encounter rate models !!
What does this mean for encounter rates ?
vR
Gerritsen & Strickler 1977
Rothschild & Osborn 1988
Evans 1989
Randomly directed prey swimming
ambushnon turbulent
2
2( )
4
Rf r
r
22
24
4
Rdn Cr dr
r
2dnZ C R v
dt
Encounter rate:
Ballistic model
uRCZ 2
Ballistic model variations
Gerritsen & Strickler 1977
u
vuRCZ
3
3 222
Ballistic model variations
moving prey
2/122
2222
)(3
)43(
wu
wvuRCZ
Rothschild & Osborn 1988
Ballistic model variations
turbulence
2/12222 )2( wvuRCZ
Evans 1989
Ballistic model variations
gaussian distribution
Z(t) = 4/3 C R v
24 3( ) 1
3
RZ t CRv
v t
Random walk diffusion
3
vD
Diffusion equation
R
22
1c cr D
t r r r
c(R) = 0, c(r) = C
ˆ ˆ( )S
cZ t D d
r
r s
time dependent
steady state
Encounter rate:
Diffusive model
2Z C R vBallistic
4
3Z CRv Diffusive
quantitatively and qualitatively different
asymptotic limits
scale of the process under consideration
Two encounter rate models
Case a
Case b
/R = 0.1
/R = 2
Numerical simulation: basic setup
Swimming speed and detection distance remain the same, only the tumbling rate changes
Count the number of particles that are encountered each time step
Those that are encountered are set to inactive
Cyclic boundary conditions
Reactivated when crossing boundaries
t (s)
0 1 2 3 4 50
1
2
3
4
5
6
t (s)
0 1 2 3 4 50
2
4
6
8
10
12
(m
m3 /
s)
ballistic
diffusive(∞)
diffusive(t)
experimental/R = 0.1
/R = 2
Visser & Kiørboe (submitted)
Cle
aran
ce r
ate
= Z
/CNumerical simulation: results
Case a
Case b
Encounter rate cannot be faster than ballistic
Ballistic – Diffusive transition (meso-diffusion) when R ≈
Maxwell – Cattaneo equations in 1 D
C J
t x
P J
J Dx t
Telegraph equationKelvin 1860's
Unsolved problem in 3 D, even for simple geometries
Consequences for plankton motility ?
Ballistic – Diffusive encounters
predator
prey
ballistic diffusive
mean path lengthRprey Rpredator< <
Is there an optimal behaviour for organisms ?
predator to prey scaling ≈ 10:1
Visser & Kiørboe (submitted)
Size of organism: esd (cm)
10-4 10-3 10-2 10-1
mot
ility
leng
th s
cale
(cm
)
10-4
10-3
10-2
10-1
100
Marine Bacterium TW-3
Microscilla furvescens
Bodo designis
Spumella sp.
Herterocapsa triquetra
Balanion comatum
Acartia tonsa
Centropages typicus
Temora longicornis
Calanus helgolandicus
protists
copepod nauplii
bacteria
adultcopepods
= 7 d (r2 = 0.90)
Motility length vrs body size
Summary statements
Random walk models are an simple way to link the behaviour of individuals to macroscopic effects at the population and environmental level.
Continuous random walks appear to be a good model for the motility of plankton.
Ballistic – Diffusive aspects of encounter processes are important.
The wrong model can give wildly different estimates of encounter rate.
Unanswered problem in classical physics
Top down control on planktonic motility patterns ?