5.2 Entrainment

Question 1

Define complexity

Answer 1

Emergent behaviour fo systems can’t be obtained by the behvaiour of individual elements

Question 2

Describe how compeititon can contriubute to disorder

Answer 2

There is a competition of order (like neighbour) and disorder (different to neighbour)

Question 3

Describe the order/disorder of a ferromagnet

Answer 3

Order - Spins align

Disorder - Thermal fluctuations

Question 4

Describe the order and control parameters of the ferromagnet

Answer 4

Order - Magnetisation

Control - Temperature (fluctuations) vs strength of spin-spin couping

Question 5

What is the macroscopic behaviour / order-disorder transition for the ferromagnet?

Answer 5

That the net magnetisaion can be 0 or no zero due to alignment/disalignment of spins

Question 6

State and explain an entrainment example

Answer 6

N disordered tourists

Or ordered tourists that follow the tour guide

Question 7

What is the order parameter of the tourist example?

Answer 7

The average speed of the tourists which is determined by the tour guide

Question 8

Describe why entrained motion is desirable

Answer 8

Just one equation to solve rather than N

e.g. tour guide rather than tourists

Question 9

Describe how we can integrate an equation using the integrating factor

Answer 9

Need equation of the the form:
dy/dt + f (t) y = g(t)
Multiply all terms by IF = exp [ ∫ f (t) dt ] and you get a perfect integral

Question 10

State the generalised differential equation for entrainment and define all terms

Answer 10

dq_i / dt = H_i (q_j) + S_i (t)
H_i is a function of q_i, and S_i is the external driving term
Assume q_i interact on timescales &laquo_space;S_i

Question 11

How does the generalised differential equation change for self organisation?

Answer 11

dq_i / dt = H_i (q_j) = -α_i q_i + G(q_c)

q_c = sum over q_j - Centre of mass of system