cosmicmolecularquanta does the inertia of a body depend on its energy content? annalen der physik,...

Cosmic Molecular Quanta

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 ?