A BOLTZMANN MODEL FOR SCHOOLING OF FISH Maria C. Carvalho University of Lisbon Joint work with E. Carlen, P . Degond and B. Wennberg A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 1/49
The kinetic equation The main concern of this paper is the following Boltzmann equation: � � π � π f ( t, x ′ ) f ( t, x ′ + y ) g ( x − x ′ − y ∂ t f ( t, x ) = 2) − π − π � β ( | sin( y/ 2) | ) dx ′ dy − f ( t, x ) f ( t, x + y ) 2 π . (1) 2 π The unknown f is a probability density on T 1 , giving e.g. the distribution of directions in a fish school, and g is a given probability density modeling the noise in the model. In our case β is just a constant or, perhaps more realistically, β ( x ) = | x | . A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 2/49
The microscopic model This kinetic equation has been rigorously derived as a limit as N → ∞ of an pair interaction driven N -particle system . These systems are defined as Markov jump processes in an N -fold product space T N , where jumps only involve two coordinates. The jumps are triggered by a Poisson clock with a rate proportional to N , and the outcome of the jump is independent of the clock. A jump involves first a choice of a pair ( j, k ) from the set 1 ≤ j < k ≤ N , and then a transition x �→ x ′ , independent of ( j, k ) : x = ( x 1 , ...., x j , ..., x k , ...., x N ) �→ k , ...., x N ) = x ′ . ( x 1 , ...., x ′ j , ..., x ′ (2) A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 3/49
This jump process we study here has the following state space and jump rules. The state space is the N -dimensional torus. We use coordinates z j = exp( ix j ) ∈ C . The jumps take a pair ( z j , z k ) to z j,k e iX j , � z j,k e iX k ) , ( z ′ j , z ′ k ) = ( � (3) where z j,k = z j + z k � | z j + z k | is the mid point of the smallest interval on the circle limited by z j and z k , and X j and X k are independent and equally distributed angles. A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 4/49
z jk � z j z ′ j z ′ k z k z jk = z j + z k Figure 1: � | z j + z k | A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 5/49
It is clear from this picture the process produces correlations. It is also clear that to describe schooling of fish, the process must produce correlations While this may seem inconsistent with propagation of chaos, it has been shown that propagation of chaos does hold for this model, and the kinetic equation we study has been rigorously derived from it (Carlen, Degond and Wennberg 2013). A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 6/49
Our focus here is on the steady states of this equation. While the uniform distribution is always a steady state, no matter what the noise is, it is not always stable. Our main result is a proof of existence of stable, sharply peaked steady states. A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 7/49
Fourier variables Because all functions are periodic, it is natural to consider to rewrite the system in terms of the Fourier series. Multiplying our kinetic equation with a test function ϕ , integrating over [ − π, π ] , and performing a change of variables gives � d S 1 f ( t, x ) ϕ ( x ) dx 2 π = dt � β ( y ) ( ϕ ( x + y/ 2 + z ) − ϕ ( x )) dx dy dz − π,π 3 f ( t, x ) f ( t, x + y ) g ( z )˜ 2 π , 2 π 2 π where ˜ β = 1 (the Maxwellian case) or ˜ β = | sin( y/ 2) | (the “hard sphere“ case). A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 8/49
Introduce � π f ( t, x ) e − ikx dx a k ( t ) = 2 π . − π � π (2 π ) − 1 g ( x ) e − ikx dx, γ k = − π � π ˜ (2 π ) − 1 β ( y ) e iuy dy, Γ( u ) = − π A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 9/49
Consider the case in which g and f , are even, so that γ k = γ − k and a k = a − k . Then they can be written as cosine-series, ∞ � g ( x ) = 1 + 2 γ k cos( kx ) , k =1 ∞ � f ( t, x ) = 1 + 2 a k ( t ) cos( kx ) . k =1 Solving for the a k ( t ) gives us the evolution of f ( t, x ) . A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 10/49
Now, taking the choice ϕ ( x ) = e − ikx in the equation � d S 1 f ( t, x ) ϕ ( x ) dx 2 π = dt � β ( y ) ( ϕ ( x + y/ 2 + z ) − ϕ ( x )) dx dy dz [ − π,π ] 3 f ( t, x ) f ( t, x + y ) g ( z )˜ 2 π , 2 π 2 π and doing the integrals, one gets an evolution equation for the Fourier coefficients: A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 11/49
Proposition 0.1. Suppose that g is even. Let a k ( t ) be the Fourier coefficients of a solution of our evolution equation which is an even probability density. Then, a 0 = 1 and a k for k � = 0 satisfy a − k = a k and solve the following system: d dta k ( t ) = ( 2 γ k Γ( k/ 2) − Γ(0) − Γ( k ) ) a k ( t ) + k − 1 � ( γ k Γ( n − k/ 2) − Γ( n )) a n ( t ) a k − n ( t ) + n =1 ∞ � (2 γ k Γ( n − k/ 2) − Γ( n ) − Γ( n − k )) a n ( t ) a n − k ( t ) n = k +1 A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 12/49
The function Γ( u ) , which is to be evaluated only on half-integer points, is Γ( u ) = sin( πu ) πu which is 1 when u = 0 0 when u ∈ Z \ { 0 } 2( − 1) ℓ when u = ℓ + 1 / 2 π (2 ℓ + 1) in the Maxwellian case, which is the case ˜ β (1) ≡ 1 . A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 13/49
We will mostly be concerned with the Maxwellian case, but the equations for the Fourier coefficients can be worked out also in the hard-sphere case, when ˜ β ( y ) = | sin( y/ 2) | . In this case, Γ( u ) = 2 − 4 u sin( πu ) π − 4 πu 2 which is 2 / ( π (1 − 4 u 2 )) when u ∈ Z when 1 /π u = ± 1 / 2 , 2( − 1) ℓ ℓ + ( − 1) ℓ − 1 when u = ℓ + 1 / 2 , ℓ � = 0 , − 1 2 πℓ 2 + 2 πℓ A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 14/49
Finitely many modes in the noise Notice that since Γ( j ) = 0 for all j ∈ N , if it is the case that � π g ( x ) e − ikx dx γ k := � g ( k ) = 2 π = 0 , − π the equation for a k ( t ) reduces to d dta k ( t ) = − Γ(0) a k ( t ) = − a k ( t ) The important conclusion is that if the noise distribution g has only finitely many Fourier modes, so that for some integer N , γ k = 0 for | k | > N , then for all k such that | k | > N , a k ( t ) tends to zero exponentially fast. Only the modes with | k | ≤ N can be present in a steady state. A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 15/49
Now we rewrite our evolution equation, which is d dta k ( t ) = ( 2 γ k Γ( k/ 2) − Γ(0) − Γ( k ) ) a k ( t ) + k − 1 � ( γ k Γ( n − k/ 2) − Γ( n )) a n ( t ) a k − n ( t ) + n =1 � ∞ (2 γ k Γ( n − k/ 2) − Γ( n ) − Γ( n − k )) a n ( t ) a n − k ( t ) n = k +1 in a manner to make this evident, specializing to the Maxwellian case A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 16/49
The Maxwellian case equation Finally our equation for the Maxwellian case is d dta k ( t ) = ( 2 γ k Γ( k/ 2) − 1 ) a k ( t ) + k − 1 � γ k Γ( n − k/ 2) a n ( t ) a k − n ( t ) + n =1 ∞ � 2 γ k Γ( n − k/ 2) a n ( t ) a n − k ( t ) (4) n = k +1 We now have our equation in a form that is adapted to the study of stability and existence of steady states. We begin with the stability. A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 17/49
The stability analysis To investigate the stability of the uniform density, let f ( x, t ) = 1 + εF ( x, t ) , and let b k ( t ) , k ∈ Z be the Fourier coefficients of F ( x, t ) . Then b 0 = 0 , and for k � = 0 , d dtb k ( t ) = b k ( t ) (2 γ k Γ( k/ 2) − Γ(0) − Γ( k )) + O ( ε ) Hence the linearized stability may be determined by analysing separately λ k = (2 γ k Γ( k/ 2) − Γ(0) − Γ( k )) . Thus each Fourier mode is an eigenfunction of the linearized operator and for k � = 0 the eigenvalue is λ k = 2 γ k Γ( k/ 2) − Γ(0) − Γ( k ) . A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 18/49
In the Maxwellian case. Γ(0) + Γ( k ) = 1 for all positive integers k , and � kπ � Γ( k/ 2) = 2 sin 2 . kπ Therefore, � kπ � 4 sin 2 λ k = γ k − 1 . kπ Whatever the choice of g , | γ k | ≤ 1 , so that k > 1 ⇒ λ k < 0 . A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 19/49
This leads to: Theorem 0.2. In the Maxwellian case, there is at most one linearly unstable mode at the uniform density, namely the k = 1 mode. The uniform density is linearly stable if and only if g (1) ≤ π γ 1 := � 4 . ( ∗ ) When there is strict inequality in ( ∗ ) , the uniform density is (non-linearly) stable. This theorem points to the special role that a 1 := � f (1) will play in the proof of existence of nonuniform steady states. We would expect that whenever the noise function g is such that ( ∗ ) is violated, there will exist a stable, non-uniform steady state. A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 20/49
An example Before turning to the existence of stable non-uniform steady states, let us look at a concrete example. Consider a family of distributions g ( y ) defined as the periodization of 1 τ ρ ( y τ ) , where ρ is a given, even, probability density on R : � y − 2 πj � ∞ � 1 g τ ( y ) = 2 π τ ρ τ j = −∞ Then: A BOLTZMANN MODEL FOR SCHOOLING OF FISH – p. 21/49
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