Upload
others
View
8
Download
0
Embed Size (px)
Citation preview
Symmetry, Functional Analisis and Non-linearPhysics
Renato Alvarez-Nodarse
IMUS & Dpto. Analisis Matematico, Universidad de Sevilla
v ∼ ǫ21ǫ2 cos(φ+ θ)
E.I.T.A. 2014, Alquezar, 17-19 de octubre de 2014
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Centenario del nacimiento de Luis Vigil
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
El Teorema de Galileo
Galileo: La filosofıa [natural]esta escrita en ese grandioso libroque tenemos abierto ante los ojos,(quiero decir, el universo), perono se puede entender si antes nose aprende a entender la lengua, aconocer los caracteres en los queesta escrito. Esta escrito en lenguamatematica y sus caracteres sontriangulos, cırculos y otras figurasgeometricas, sin las cuales esimposible entender ni una palabra;sin ellos es como girar vanamenteen un oscuro laberinto.
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
El Teorema de Galileo
Galileo: La filosofıa [natural]esta escrita en ese grandioso libroque tenemos abierto ante los ojos,(quiero decir, el universo), perono se puede entender si antes nose aprende a entender la lengua, aconocer los caracteres en los queesta escrito. Esta escrito en lenguamatematica y sus caracteres sontriangulos, cırculos y otras figurasgeometricas, sin las cuales esimposible entender ni una palabra;sin ellos es como girar vanamenteen un oscuro laberinto.
⇒ Corolario: Si queremos conocer el mundo que nos rodeatenemos que saber Matematicas.
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Invariance (Symmetry)
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Invariance (Symmetry)
Invariance: a thing is symmetric if there is something we can
do to it so that after we have done it, it looks the same as it did
before.
Let us consider a .
Transformations: We can do thing to our : T : 7→ ′: To donothing, cut it, compress it, deform it, rotate it ...
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Invariance (Symmetry)
Invariance: a thing is symmetric if there is something we can
do to it so that after we have done it, it looks the same as it did
before.
Let us consider a .
Transformations: We can do thing to our : T : 7→ ′: To donothing, cut it, compress it, deform it, rotate it ...
Invariance: The way the new ′ looks should be identical to theoriginal
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Playing with the symmetry
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Playing with the symmetry
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Playing with the symmetry
A
B
C
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Playing with the symmetry
B
C
A
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Playing with the symmetry
A
B
C
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Playing with the symmetry
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A very important Symmetry: The time-shift symmetry
What time-shift invariant of the measurement means?
Time-shift invariant: The experiments can be repeated at anytime. Under the same conditions, the same results are obtained.
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Motivation: A problem from Physics
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!
system
π
−3.12.3
System: device, equations, experiments, simulations...
Input: harmonic (sinusoidal) functions
Output: a number, which is an average measurement
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Motivation: A problem from Physics
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!
system
π
−3.12.3
System: device, equations, experiments, simulations...
Input: harmonic (sinusoidal) functions
Output: a number, which is an average measurement
It has been found in experiments, simulations, . . . that theoutput has the same functional dependence on the parameters ofthe input
Why?
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Math. formulation of the problem
The output is a functional of an arbitrary number, n, of externalforces fi (t), i = 1, 2, . . . , n
In the following we will denote the output (functional) byv [f(t)], and the input will the the vector f(t).
An special emphasis will be done when fi (t) = ǫi cos(qiωt + φi ),i = 1, 2, . . . , n )
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Math. formulation of the problem
The output is a functional of an arbitrary number, n, of externalforces fi (t), i = 1, 2, . . . , n
In the following we will denote the output (functional) byv [f(t)], and the input will the the vector f(t).
An special emphasis will be done when fi (t) = ǫi cos(qiωt + φi ),i = 1, 2, . . . , n )
v [f] is time-shift invariant functional, i.e., for any τ
v [f(t)] = v [f(t + τ)]
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Math. formulation of the problem
The output is a functional of an arbitrary number, n, of externalforces fi (t), i = 1, 2, . . . , n
In the following we will denote the output (functional) byv [f(t)], and the input will the the vector f(t).
An special emphasis will be done when fi (t) = ǫi cos(qiωt + φi ),i = 1, 2, . . . , n )
v [f] is time-shift invariant functional, i.e., for any τ
v [f(t)] = v [f(t + τ)]
Implications in Physics: the experiments can be repeated at anytime. Under the same conditions, the same results are obtained.
Implications in Mathematics: the dependence of v of theamplitudes and phases of f are fixed and can be calculated as longas v is sufficiently smooth.
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Paradigmatic example: The ratchet effect
Ratchet effect: net motion of particles or solitons induced byperiodic or random, zero-average external forces f.
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Paradigmatic example: The ratchet effect
Ratchet effect: net motion of particles or solitons induced byperiodic or random, zero-average external forces f.
The motion of cold atoms in symmetric optical lattices, (Schiavoni
et al. PRL 2003; Gommers, et al. PRL 2005; PRL 2006; PRA 2007; PRL 2008). The motion offluxons in the Josephson junctions (Ustinov et al. PRL 2004; Ooi et al. PRL 2007).
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Paradigmatic example: The ratchet effect
Ratchet effect: net motion of particles or solitons induced byperiodic or random, zero-average external forces f.
The motion of cold atoms in symmetric optical lattices, (Schiavoni
et al. PRL 2003; Gommers, et al. PRL 2005; PRL 2006; PRA 2007; PRL 2008). The motion offluxons in the Josephson junctions (Ustinov et al. PRL 2004; Ooi et al. PRL 2007).
Measurement of x(t) (position of particles or non-linear waves)
Ratchet velocity v 6= 0 is an average velocity induced by 〈f〉 = 0
v := v [f(t)] = lımt→+∞
1
t
∫ t
0x(τ) dτ = lım
t→+∞
x(t)− x(0)
t.
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Universality: “Same cause” ⇒ “same effect”
A fact: Very different non-linear systems driven by the forcef (t) = ǫ1 cos(pΩt) + ǫ2 cos(qΩt + φ)
Experiments: optical lattices, ferrofluids, semiconductors,Josephson junctions . . .
Simulations: dissipative dynamics of particles in symmetricpotentials
Simulations: dissipative dynamics of non-linear Schrodingersolitons in symmetric potentials
Theory: dissipative dynamics of sine-Gordon and φ4 solitons
Theory: overdamped dynamics of particles in symmetricpotentials
have the same behaviour:
v [f(t)] = Aǫp1ǫq2 cos(pφ+ θ0)
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Universality: “Same cause” ⇒ “same effect”
A fact: Very different non-linear systems driven by the forcef (t) = ǫ1 cos(pΩt) + ǫ2 cos(qΩt + φ)
Experiments: optical lattices, ferrofluids, semiconductors,Josephson junctions . . .
Simulations: dissipative dynamics of particles in symmetricpotentials
Simulations: dissipative dynamics of non-linear Schrodingersolitons in symmetric potentials
Theory: dissipative dynamics of sine-Gordon and φ4 solitons
Theory: overdamped dynamics of particles in symmetricpotentials
have the same behaviour:
v [f(t)] = Aǫp1ǫq2 cos(pφ+ θ0)
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A special case: f (t) = ǫ1 cos(Ωt) + ǫ2 cos(2Ωt + φ)
A first attempt for understanding this: The method of moments . . .
which is based on the assumption that v [f(t)] can be spanned asthe Taylor series
x(t) =∞∑
k=1
ck(f (t))2k+1 ⇒
v [f(t)] = 〈x〉 =∞∑
k=1
c2k+1〈(f (t))2k+1〉,
where here, we use the notation 〈X 〉 := 1/T∫ T
0 X (t) dt.
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A special case: f (t) = ǫ1 cos(Ωt) + ǫ2 cos(2Ωt + φ)
A first attempt for understanding this: The method of moments . . .
which is based on the assumption that v [f(t)] can be spanned asthe Taylor series
x(t) =∞∑
k=1
ck(f (t))2k+1 ⇒
v [f(t)] = 〈x〉 =∞∑
k=1
c2k+1〈(f (t))2k+1〉,
where here, we use the notation 〈X 〉 := 1/T∫ T
0 X (t) dt.
First approximation: v [f(t)] = A〈(f (t))3〉 = Aǫ21ǫ2 cos(φ)
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A special case: f (t) = ǫ1 cos(Ωt) + ǫ2 cos(2Ωt + φ)
A first attempt for understanding this: The method of moments . . .
which is based on the assumption that v [f(t)] can be spanned asthe Taylor series
x(t) =∞∑
k=1
ck(f (t))2k+1 ⇒
v [f(t)] = 〈x〉 =∞∑
k=1
c2k+1〈(f (t))2k+1〉,
where here, we use the notation 〈X 〉 := 1/T∫ T
0 X (t) dt.
First approximation: v [f(t)] = A〈(f (t))3〉 = Aǫ21ǫ2 cos(φ)
What happens with the phase! v [f(t)] = Aǫ21ǫ2 cos(φ+ θ0)
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Why the methods of moments does not work?
Because the assumption that v [f(t)] can be spanned as Taylorseries is wrong: v [f(t)] is a functional, thus we should use afunctional series expansion: f(t) = (f1(t), f2(t))
v [f] =
∞∑
n1=0
∞∑
n2=0
〈cn1,n2(t11, · · · , t1n1 , t21, . . . , t2n2)
× f1(t11) · · · f1(t1n1)f2(t21) · · · f2(t2n2)〉,
where the kernels cn1,n2 are real, T -periodic and symmetric. Here
〈Ω(t1, . . . , tr )〉 =1
T r
∫ T
0dt1 · · ·
∫ T
0dtr Ω(t1, . . . , tr )
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
But the situation is worst ...
The method of moments is wrong. Just one example: Let be asystem defined by (x is the position, u is the velocity)
d
dt
Mu√1− u2
= −f (t)− γu√
1− u2,
dx
dt= u(t),
f (t) = ǫ1 cos(ωt+φ1)+ǫ2 cos(2ωt+φ2), u(0) = u0, x(0) = x0,
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
But the situation is worst ...
The method of moments is wrong. Just one example: Let be asystem defined by (x is the position, u is the velocity)
d
dt
Mu√1− u2
= −f (t)− γu√
1− u2,
dx
dt= u(t),
f (t) = ǫ1 cos(ωt+φ1)+ǫ2 cos(2ωt+φ2), u(0) = u0, x(0) = x0,
It has an exact solution: u(t) =
∞∑
k=0
(−1)k(1/2)kk!M2k+1
[P(t)]2k+1
donde P(t) = · · · and v = Bǫ21ǫ2 cos(2φ1 − φ2 + θ0︸︷︷︸
6=0
).
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
But the situation is worst ...
The method of moments is wrong. Just one example: Let be asystem defined by (x is the position, u is the velocity)
d
dt
Mu√1− u2
= −f (t)− γu√
1− u2,
dx
dt= u(t),
f (t) = ǫ1 cos(ωt+φ1)+ǫ2 cos(2ωt+φ2), u(0) = u0, x(0) = x0,
It has an exact solution: u(t) =
∞∑
k=0
(−1)k(1/2)kk!M2k+1
[P(t)]2k+1
donde P(t) = · · · and v = Bǫ21ǫ2 cos(2φ1 − φ2 + θ0︸︷︷︸
6=0
).
If we use the force f (t) =
ǫ1 if 0< t<Tl
0 if Tl < t<T−Tl
−ǫ1 if T−Tl < t<T
⇒
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
But the situation is worst ...
The method of moments is wrong. Just one example: Let be asystem defined by (x is the position, u is the velocity)
d
dt
Mu√1− u2
= −f (t)− γu√
1− u2,
dx
dt= u(t),
f (t) = ǫ1 cos(ωt+φ1)+ǫ2 cos(2ωt+φ2), u(0) = u0, x(0) = x0,
It has an exact solution: u(t) =
∞∑
k=0
(−1)k(1/2)kk!M2k+1
[P(t)]2k+1
donde P(t) = · · · and v = Bǫ21ǫ2 cos(2φ1 − φ2 + θ0︸︷︷︸
6=0
).
If we use the force f (t) =
ǫ1 if 0< t<Tl
0 if Tl < t<T−Tl
−ǫ1 if T−Tl < t<T
⇒〈f 2k+1〉 = 0
⇓v = 0
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A little math ...
We recall that v [f] not an arbitrary functional. It is
a time-shift invariant functional, ∀τ
v [f(t)] = v [f(t + τ)]
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A little math ...
We recall that v [f] not an arbitrary functional. It is
a time-shift invariant functional, ∀τ
v [f(t)] = v [f(t + τ)]
How the kernels are affected by this symmetry?
v [f] =
∞∑
n1=0
∞∑
n2=0
〈cn1,n2(t11, · · · , t1n1 , t21, . . . , t2n2)
× f1(t11) · · · f1(t1n1)f2(t21) · · · f2(t2n2)〉.
⇓
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A little math ...
We recall that v [f] not an arbitrary functional. It is
a time-shift invariant functional, ∀τ
v [f(t)] = v [f(t + τ)]
How the kernels are affected by this symmetry?
v [f] =
∞∑
n1=0
∞∑
n2=0
〈cn1,n2(t11, · · · , t1n1 , t21, . . . , t2n2)
× f1(t11) · · · f1(t1n1)f2(t21) · · · f2(t2n2)〉.
⇓Fundamental property ∀τ ∈ (0,T ):
cn1,n2(t11−τ, · · · , t1n1−τ, t21−τ, . . . , t2n2−τ) = cn1,n2(t11, · · · , t1n1 , t21, . . . , t2n2)
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Functional Taylor expansion of time-shift invariant functional v [f1, f2, · · · ]
Theorem: Let f(t) = ǫ1 cos(q1ωt + φ1), ǫ2 cos(q2ωt + φ2).If ∀τ v [f(t + τ)] = v [f(t)] and v [f] smooth enough then
v [f1, f2] = C0(ǫ1, ǫ2) +∑
x∈D+
ǫ|x1|1 ǫ
|x2|2 Cx(ǫ1, ǫ2) cos(x ·φ+ θx(ǫ1, ǫ2)).
x := (x1, x2) ∈ D+: set of the solutions of the DiophantineEq. q1x1 + q2x2 = 0 with x1 > 0.
φ = (φ1, φ2).
Cx and θx are functions of ǫ21 and ǫ22.
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Functional Taylor expansion of time-shift invariant functional v [f1, f2, · · · ]
Theorem: Let f(t) = ǫ1 cos(q1ωt + φ1), ǫ2 cos(q2ωt + φ2).If ∀τ v [f(t + τ)] = v [f(t)] and v [f] smooth enough then
v [f1, f2] = C0(ǫ1, ǫ2) +∑
x∈D+
ǫ|x1|1 ǫ
|x2|2 Cx(ǫ1, ǫ2) cos(x ·φ+ θx(ǫ1, ǫ2)).
x := (x1, x2) ∈ D+: set of the solutions of the DiophantineEq. q1x1 + q2x2 = 0 with x1 > 0.
φ = (φ1, φ2).
Cx and θx are functions of ǫ21 and ǫ22.
Equivalently
v=C0(ǫ1, ǫ2)+∞∑
k=1
(ǫq21 ǫq12 )kCk(ǫ1, ǫ2) cos(k(q1φ2−q2φ1)+θk(ǫ1, ǫ2))
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Theory vs. Experiments: going down to the Earth
small amplitude limit: v = C0ǫ21ǫ2 cos(φ+ θ0), agrees with the
simulations and experiments e.g. optical lattices, bi-harmonic force.
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Theory vs. Experiments: going down to the Earth
small amplitude limit: v = C0ǫ21ǫ2 cos(φ+ θ0), agrees with the
simulations and experiments e.g. optical lattices, bi-harmonic force.
intermediate amplitudes: v = vmax(ǫ1, ǫ2) cos(φ+ θ1(ǫ1, ǫ2))(agrees with experiments on optical lattices)
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Theory vs. Experiments: going down to the Earth
small amplitude limit: v = C0ǫ21ǫ2 cos(φ+ θ0), agrees with the
simulations and experiments e.g. optical lattices, bi-harmonic force.
intermediate amplitudes: v = vmax(ǫ1, ǫ2) cos(φ+ θ1(ǫ1, ǫ2))(agrees with experiments on optical lattices)
large amplitudes: Prediction of the Theory: By increasing theamplitude of the forces, the direction of the motion can bechanged! This has been observed in experiments: optical lattices,bi-harmonic force. Cubero, et. al., PRE 2010; also in simulations ofthe NLKG, etc.
+ a lot of interesting Math Corollaries ...
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
An astonishing phenomena: Not math model at all!
(PRL, Noblin et. al), fv = ǫ1 cos(ωt + φ), fh = ǫ2 cos(ωt).
0 π/2 π 3π/2 2πφ
-1
-0.5
0
0.5
1
v (cm/s)
v [−fv , fh] = −v [fv , fh]
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
An astonishing phenomena: Not math model at all!
(PRL, Noblin et. al), fv = ǫ1 cos(ωt + φ), fh = ǫ2 cos(ωt).
0 π/2 π 3π/2 2πφ
-1
-0.5
0
0.5
1
v (cm/s)
v [−fv , fh] = −v [fv , fh]
v = vmax(ǫ1, ǫ2) cos(φ+ θ1(ǫ1, ǫ2)) + v2(ǫ1, ǫ2) cos(3φ+ θ2(ǫ1, ǫ2)) + · · · large amplitude regime: v is no longer a sinusoidal function
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
An astonishing phenomena: Not math model at all!
(PRL, Noblin et. al), fv = ǫ1 cos(ωt + φ), fh = ǫ2 cos(ωt).
0 π/2 π 3π/2 2πφ
-1
-0.5
0
0.5
1
v (cm/s)
v [−fv , fh] = −v [fv , fh]
v = vmax(ǫ1, ǫ2) cos(φ+ θ1(ǫ1, ǫ2)) + v2(ǫ1, ǫ2) cos(3φ+ θ2(ǫ1, ǫ2)) + · · · large amplitude regime: v is no longer a sinusoidal function
Theory vs experiment: v = a cos(φ+ b) + c cos(3φ+ d)
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A curious phenomena: Could dissipation enhance the transport?
v =∞∑
k=1(odd)
(ǫq21 ǫq12 )kCk(ǫ1, ǫ2) cos(kq1φ+ θk(ǫ1, ǫ2)).
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A curious phenomena: Could dissipation enhance the transport?
v =∞∑
k=1(odd)
(ǫq21 ǫq12 )kCk(ǫ1, ǫ2) cos(kq1φ+ θk(ǫ1, ǫ2)).
Time-reversal symmetry v [f1(−t), f2(−t)] = −v [f1(t), f2(t)]⇐⇒ θk = ±π/2 (Hamiltonian limit)
Time-reversal symmetry v [f1(−t), f2(−t)] = v [f1(t), f2(t)] ⇐⇒θk = 0 or π (Overdamped systems)
Time-reversal symmetries fix the “phase lag”
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A curious phenomena: Could dissipation enhance the transport?
v =∞∑
k=1(odd)
(ǫq21 ǫq12 )kCk(ǫ1, ǫ2) cos(kq1φ+ θk(ǫ1, ǫ2)).
Time-reversal symmetry v [f1(−t), f2(−t)] = −v [f1(t), f2(t)]⇐⇒ θk = ±π/2 (Hamiltonian limit)
Time-reversal symmetry v [f1(−t), f2(−t)] = v [f1(t), f2(t)] ⇐⇒θk = 0 or π (Overdamped systems)
Time-reversal symmetries fix the “phase lag”
By setting φ = 0, close to the Hamiltonian limit, v = 0. Byincreasing the damping, the transport is induced. We have beensee this in the equation for a relativistic particle discussed before!
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Main ingredients of the proof: fi = ǫi cos(qiωt + φi)
v [f] =
∞∑
n1=0
∞∑
n2=0
〈cn1,n2(t11, · · · , t1n1 , t21, . . . , t2n2)
× f1(t11) · · · f1(t1n1)f2(t21) · · · f2(t2n2)〉,
∀τ, v [f(t + τ)] = v [f(t)] ⇒ cn(t1+ τ, · · · , tn + τ) = cn(t1, · · · , tn)
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Main ingredients of the proof: fi = ǫi cos(qiωt + φi)
v [f] =
∞∑
n1=0
∞∑
n2=0
〈cn1,n2(t11, · · · , t1n1 , t21, . . . , t2n2)
× f1(t11) · · · f1(t1n1)f2(t21) · · · f2(t2n2)〉,
∀τ, v [f(t + τ)] = v [f(t)] ⇒ cn(t1+ τ, · · · , tn + τ) = cn(t1, · · · , tn)
v [f] =∑
k,l∈N20
ǫk1+l11 ǫk2+l2
2 e i(k−l)·φA(k, l),
time-shift invariace ⇒ A(k, l) = 0 if q · (k− l) 6= 0
q · (k− l) = 0 Diophantine Eq.
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Main ingredients of the proof: fi = ǫi cos(qiωt + φi)
v [f] =
∞∑
n1=0
∞∑
n2=0
〈cn1,n2(t11, · · · , t1n1 , t21, . . . , t2n2)
× f1(t11) · · · f1(t1n1)f2(t21) · · · f2(t2n2)〉,
∀τ, v [f(t + τ)] = v [f(t)] ⇒ cn(t1+ τ, · · · , tn + τ) = cn(t1, · · · , tn)
v [f] =∑
k,l∈N20
ǫk1+l11 ǫk2+l2
2 e i(k−l)·φA(k, l),
time-shift invariace ⇒ A(k, l) = 0 if q · (k− l) 6= 0
q · (k− l) = 0 Diophantine Eq.
The solutions of the Eq. q · (k− l) = 0 determine k, l
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Main ingredients of the proof: fi = ǫi cos(qiωt + φi)
v [f] =
∞∑
n1=0
∞∑
n2=0
〈cn1,n2(t11, · · · , t1n1 , t21, . . . , t2n2)
× f1(t11) · · · f1(t1n1)f2(t21) · · · f2(t2n2)〉,
∀τ, v [f(t + τ)] = v [f(t)] ⇒ cn(t1+ τ, · · · , tn + τ) = cn(t1, · · · , tn)
v [f] =∑
k,l∈N20
ǫk1+l11 ǫk2+l2
2 e i(k−l)·φA(k, l),
time-shift invariace ⇒ A(k, l) = 0 if q · (k− l) 6= 0
q · (k− l) = 0 Diophantine Eq.
The solutions of the Eq. q · (k− l) = 0 determine k, l
For arbitrary n see (Cuesta, Quintero, Renato, PRX 2013)Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A simple proof ... Why another proof?
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A simple proof ... Why another proof?
Input: fj(t) = ǫj cos [ωj(t − t0) + ϕj ]
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A simple proof ... Why another proof?
Input: fj(t) = ǫj cos [ωj(t − t0) + ϕj ]
This set of periodic functions is invariant under these twotransformations
T1 : t0,ω,ϕ, ǫ 7−→ t0,ω,ϕ+ π (j), ǫ(j) ,T2 : t0,ω,ϕ, ǫ 7−→ 0,ω,ϕ− t0ω, ǫ ,
where
(j) canonical basis of Rn and the vector ǫ(j) is obtainedfrom the vector ǫ by replacing its jth component by −ǫj .
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A simple proof ... Why another proof?
Input: fj(t) = ǫj cos [ωj(t − t0) + ϕj ]
This set of periodic functions is invariant under these twotransformations
T1 : t0,ω,ϕ, ǫ 7−→ t0,ω,ϕ+ π (j), ǫ(j) ,T2 : t0,ω,ϕ, ǫ 7−→ 0,ω,ϕ− t0ω, ǫ ,
where
(j) canonical basis of Rn and the vector ǫ(j) is obtainedfrom the vector ǫ by replacing its jth component by −ǫj .Let Υ represent a certain (physical) quantity of the system:
Υ(t0,ϕ, ǫ) = Υ(t0,ϕ+ π (j), ǫ(j)) = Υ(0,ϕ− ωt0, ǫ) ,
By applying the first equality in the above equation twice, we seethat Υ is periodic with respect to all the components of the vectorϕ with period 2π.
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A simple proof ... Υ can be expanded in Fourier series as
Υ(ζ0,ϕ, ǫ) =∑
∈ZN
υ
(ǫ)e i(ϕ−Ωζ0)· (in real form)
! "
! "
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A simple proof ... Υ can be expanded in Fourier series as
Υ(ζ0,ϕ, ǫ) =∑
∈ZN
υ
(ǫ)e i(ϕ−Ωζ0)· (in real form)
=∑
∈ZN
c
(ǫ) cos [(ϕ− ωt0) · + χ
(ǫ)] , (1)
If Υ is time-shift invariant ⇒ ω · = 0.
! "
! "
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A simple proof ... Υ can be expanded in Fourier series as
Υ(ζ0,ϕ, ǫ) =∑
∈ZN
υ
(ǫ)e i(ϕ−Ωζ0)· (in real form)
=∑
∈ZN
c
(ǫ) cos [(ϕ− ωt0) · + χ
(ǫ)] , (1)
If Υ is time-shift invariant ⇒ ω · = 0.If ω = (q1ω, · · · , qsω) weobtain the Diophantine Eq.
! "
! "
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
A simple proof ... Υ can be expanded in Fourier series as
Υ(ζ0,ϕ, ǫ) =∑
∈ZN
υ
(ǫ)e i(ϕ−Ωζ0)· (in real form)
=∑
∈ZN
c
(ǫ) cos [(ϕ− ωt0) · + χ
(ǫ)] , (1)
If Υ is time-shift invariant ⇒ ω · = 0.If ω = (q1ω, · · · , qsω) weobtain the Diophantine Eq.
Case t0 = 0: fj(t) = ǫj cos [ωj t + ϕj ] = αj cos(ωj t) + βj sin(ωj t)
and analytic output: Υ =∑
! ,"∈NN0a! ,"
∏Nj=1 α
ljj β
rjj , then computing
the Fourier coeff.
υ
(ǫ) =
∫ π
−π. . .
∫ π
−π
dNϕ
(2π)NΥ(0,ϕ, ǫ)e−iϕ· .
givesRenato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Υ can be expanded as the following series
Υ(ϕ, ǫ) =∑
k∈D+
|γ
(ǫ)|
N∏
j=1
ǫ|kj |j
cos [ϕ · + χ
(ǫ)] ,
γ
(ǫ) =∑
!∈NN0
b ,!
N∏
j=1
ǫ2pjj
that is the same formula obtained using the functional analysis.
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics
Thank you for your attention!
Cuesta, Quintero, RAN, Physical Review X 3, 041014 (2013).
Renato Alvarez-Nodarse Symmetry, Functional Analisis and Non-linear Physics