Newtonian Mechanics from the principle of Maximum Caliber ´ ´ Diego Gonzalez, Sergio Davis, Gonzalo Gutierrez Facultad de Ciencias, Universidad de Chile
MaxEnt 2014, September 26th, 2014
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
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Outline
Motivation MaxEnt and the divergence theorem The Maximum Caliber principle Newton’s 2nd law Inertia and potential energy Conclusions
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
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Some motivation
Why worry about interpretations? Ideas in the theme of “physics from inference” Several (most?) “laws of physics” are just of statistical nature We might find those “laws of physics” applied in completely unexpected contexts
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
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Maximum Entropy For a system of continuous degrees of freedom ~x subjected to M constraints (j=1,. . . ,M), D E fj (~x ) = Fj (1) I
the most unbiased probability distribution is M X 1 λj fj (~x ) P(~x |I) = exp − Z
(2)
j=1
Note that we are omitting for simplicity a constant “complete ignorance” distribution P(~x |I0 ). The Lagrange multipliers λj can be obtained from −
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
∂ ln Z = Fj . ∂λj
(3)
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Solving for λ: the divergence theorem For an arbitrary (differentiable) distribution P(~x |I), let us compute the expectation of a divergence (of an arbitrary vector field ω ~ ) and apply the divergence theorem. We get D E D E ∇·ω ~ (~x ) = − ω ~ (~x ) · ∇ ln P(~x |I) .
(4)
For the special case of a MaxEnt distribution, D E E X D ∇·ω ~ (~x ) = λj ω ~ (~x ) · ∇fj (~x ) .
(5)
I
I
I
j
I
This gives us an alternative route to solving for the Lagrange multipliers, a linear system of equations (by choosing different fields ω ~ ). More details and implications of this relation in: ´ S. Davis and G. Gutierrez, Phys. Rev. E 86, 051136 (2012)
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
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The Maximum Caliber principle Suppose we are now making inferences about the trajectory x(t) followed by a system in time. We constrain a functional f [x(t); τ ] which is known for τ ∈ [ti , tf ]. D E f [x(t); τ ] = F (τ ), I
We maximize the caliber or path entropy Z S = − Dx(t)P[x(t)|I] ln P[x(t)|I], under the functional constraint, obtaining 1 P[x(t)|I] = exp − Z [λ(t)]
Z
!
tf
dτ λ(τ )f [x(t); τ ] ti
where λ(τ ) is now a Lagrange multiplier function. Jaynes called the quantity S the caliber of the system, as it is analogous to the cross section of a barrel (the “tube” spawned by the possible paths). ´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
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The Maximum Caliber principle
Z
tf
dτ λ(τ )f [x(t); τ ] 1 1 P[x(t)|I] = exp − A[x(t)] . Z [λ(t)] α A[x(t)] = α
ti
The action then is the “relevant quantity” for the trajectories, and 1/α its multiplier. It is just a constant with units of action introduced to make the exponent adimensional. δP δA =0⇒ = 0. δx(t) x=xcl δx(t) x=xcl
(6)
With α > 0, the most probable trajectory xcl (t) (let us call it “classical”) is the one that minimizes the action A. ´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
7 / 19
Is this related to the Feynman path integral? Feynman’s path integral formulation:
1 1 P[x(t)|I] = exp − A[x(t)] Z α Z 1 1 P(x1 (t1 ), x2 (t2 )|I) = Dx(t) exp − A[x(t)] Z α
Z K (x1 , x2 ; t1 , t2 ) ∝
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
Dx(t) exp
i S[x(t)] ~
MaxEnt 2014, September 26th, 2014
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Maximum Caliber ˙ ); τ ) the action is the time integral of a Lagrangian, If f [x(t); τ ] = f (x(τ ), x(τ Z
t2
˙ ); τ ) dτ L(x(τ ), x(τ
A=
˙ t) = αλ(t)f (x, x; ˙ t) L(x, x;
t1
For the most probable trajectory xcl (t), the Euler-Lagrange equation holds, ∂L ∂x
−
d ∂L = 0. dt ∂ x˙ x=xcl
(Euler-Lagrange equation)
We obtain an extremum principle for (classical) mechanics without invoking quantum mechanics as an underlying theory. Classical mechanics is just the most probable answer to our inference over trajectories problem.
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
9 / 19
Newton’s 2nd law Of course we could from the start take the function f to be proportional to a known Lagrangian, but . . . Could we make it emerge for simpler constraints? Consider a coordinate which describes a random walk such that 1
2
The expected size of the displacement (∆x)2 = v (t)2 (∆t)2 is known at all times (i.e. we know the magnitude of the velocity) The time-independent probability p(x) is known
It is direct to prove that d d ˙ (m(t)x(t)) = − Φ(x) . dt dx x=xcl x=xcl
(7)
where m(t) and Φ(x) are functions which impose the first and second constraint, respectively. Not only that, for the ensemble of trajectories under those general constraints, Dd dt ´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
˙ (m(t)x(t))
E I
=−
Dd E Φ(x) dx I MaxEnt 2014, September 26th, 2014
(8)
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Newton’s 2nd law
Our constraints are D E ˙ 2 = v (t)2 x(t) D EI δ(x(t) − X ) = p(X ).
(9) (10)
for all values of t. Discretizing time, E D (xk − xk −1 )2 = vk2 ∆t 2 I D E δ(xk − X ) = p(X ).
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
(11) (12)
MaxEnt 2014, September 26th, 2014
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Newton’s 2nd law Therefore the probability of the discrete-time trajectory ~x = (x1 , x2 , . . . , xN ) is given by ! Z 1 1X 2 P(~x |I) = exp − λk (xk − xk−1 ) + dX µ(X )δ(xk − X ) Z α k ! 1 1 X 2 = exp − λk (xk − xk−1 ) + µ(xk ) . Z α
(13)
k
Continuous version: Z 1 1 2 ˙ dt λ(t)x(t) + µ(x(t)) P[x(t)|I] = exp − Z α
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
(14)
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Newton’s 2nd law
P[x(t)|I] =
Z 1 1 ˙ 2 + µ(x(t)) exp − dt λ(t)x(t) Z α
(15)
This corresponds to a system with Lagrangian 1 ˙ 2 − Φ(x) m(t)x(t) 2 under the identification m(t)/2 = λ(t) and Φ(x) = −µ(x). By the Euler-Lagrange equation, ˙ t) = λ(t)x(t) ˙ 2 + µ(x) = L(x, x,
d d ˙ (m(t)x(t)) = − Φ(x) . dt dx x=xcl x=xcl
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
(16)
(17)
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How do we describe the “non-classical” trajectories? D
B[x(t)]
E
Z =
I
Dx(t)P[x(t)|I]B[x(t)]
Generalizing the vector result, D E D E ∇·ω ~ (~x ) = − ω ~ (~x ) · ∇ ln P(~x |I) , I
I
we obtain D δW [x(t)] E δx(t) D δW [x(t)] E δx(t)
=
I
I
D E δ = − W [x(t)] ln P[x(t)|I] , δx(t) I
E 1D δA E 1D ˆ t L) . W [x(t)] = W [x(t)] · (E α δx(t) I α I
ˆt G = ( ∂ − where W [x(t)] is a test functional, and E ∂x
(18)
(19)
d ∂ dt ∂ x˙ )G.
S. Davis, arXiv:1404.3249 ´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
14 / 19
Newton’s 2nd law: ensemble
D δW [x(t)] E δx(t)
=
I
E 1D ˆ t L) . W [x(t)] · (E α I
(20)
Choosing W [x(t)] to be a constant functional, 1Dˆ E Et L = 0, α I Which is nothing but Dd dt
˙ (m(t)x(t))
E I
=−
Dd E Φ(x) . dx I
(21)
This derivation without the functional identity (just using the discretized version of the trajectory and the divergence theorem) is given in: ´ ´ D. Gonzalez, S. Davis, G. Gutierrez, Foundations of Physics 44, 923 (2014).
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
15 / 19
Inertia and potential energy as emergent
Notice the identification m(t)/2 = λ(t). This gives a meaning for the mass parameter in the Lagrangian: The more informative the constraint about the magnitude of the velocity is, the more massive the “particle” is. Now notice that we also identified Φ(x) = −µ(x). This means that The more informative the constraint about the allowed positions of a particle, the more confined it is by a potential. Note that we assign a model for p(x) and this leads us to a model for Φ(x).
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
16 / 19
A slight generalization D E ˙ δ(x(t) − V ) = q(V ; t) D EI δ(x(t) − X ) = p(X ; t) I
Z Z Z ˙ P[x(t)|I] ∝ exp − dt dV δ(x(t) − V )λ(V , t) + dX δ(x(t) − X )µ(X , t) Z 1 ˙ = exp − dt [λ(x(t), t) + µ(x(t), t)] Z ˙ t) is itself the kinetic energy! (again, Now the Lagrange multiplier function λ(x, a model for the probability of the velocity leads to a model for the kinetic energy) ∂L ∂λ ∂p ∂2L = ⇒m= = ∂ x˙ ∂ x˙ ∂ x˙ ∂ x˙ 2 Taylor-expanding λ around x˙ = 0 up to second order will give you the classical (non-relativistic) kinetic energy. p=
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
17 / 19
Conclusions
A natural generalization of Maximum Entropy inference for dynamical systems is the Maximum Caliber principle. It naturally provides us with a minimum action law, if we read the “relevant quantity” as an action. Some properties can be computed in expectation over trajectories under the Maximum Caliber distribution. It leads to Newton’s second law under relatively broad assumptions
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
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Thank you for your attention! Acknowledgements: FONDECYT grant 1140514
´ ´ D. Gonzalez, S. Davis, G. Gutierrez (UChile)
MaxEnt 2014, September 26th, 2014
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