Invariant tori for the Nosé Thermostat near the High-Temperature Limit
aa r X i v : . [ m a t h . D S ] F e b INVARIANT TORI FOR THE NOS´E THERMOSTAT NEAR THEHIGH-TEMPERATURE LIMIT
LEO T. BUTLER
Abstract.
Let H ( q, p ) = p + V ( q ) be a 1-degree of freedom mechani-cal Hamiltonian with a C r periodic potential V where r >
4. The Nos´e-thermostated system associated to H is shown to have invariant tori near theinfinite temperature limit. This is shown to be true for all thermostats similarto Nos´e’s. These results complement the result of Legoll, Luskin and Moeckelwho proved the existence of such tori near the decoupling limit [4, 5]. Introduction
The computation of equilibrium statistical properties of molecular systems isof great importance to applied subjects such as biology, chemistry, computationalphysics and materials science. These equilibrium statistical properties are phasespace integrals like f = Z f ( q, p ) d µ, d µ = exp( − βH ) d p d q/Z, (1)where q is the position of the system and p is its momentum, H = H ( q, p ) is thetotal energy of the system, β = 1 /T is the reciprocal of the equilibrium temperature T and Z = Z ( β ) is a normalization constant, also called the partition function.In practice, f = f ( q, p ) is a “measurement” or “observable”, such as the positionof the first atom in the system. The computation of the integral (eq. 1) can bevery expensive, so one often wants to replace that multi-dimensional average withthe time average ˆ f = lim T →∞ T Z T f ( q ( t ) , p ( t )) d t (2)where ( q ( t ) , p ( t )) are the position and momenta of the system at time t . In principle,ˆ f depends on the initial condition ( q (0) , p (0). When, for almost all initial conditionsthe average in (eq. 2)–called a Birkhoff average–converges to f the system is ergodic .Ergodic systems have many interesting properties, but from the point-of-view here,they provide a means to an end: reduction of the multi-variable integral (eq. 1) toa single-variable integral.In equilibrium statistical mechanics, the Hamiltonian H is the internal energy ofan infinitesimal system S that is immersed in a heat bath B at the temperature T .A simple model of the exchange of energy between the infinitesimal system S and Date : September 17, 2018.2010
Mathematics Subject Classification.
Key words and phrases. thermostats; Nos´e-Hoover thermostat; Hamiltonian mechanics; KAMtheory. heat bath B was introduced by Nos´e [7]. This consists of adding an extra degreeof freedom s and rescaling momentum by s : F = H ( q, ps − ) + 12 M p s + nkT ln s | {z } N , (3)where n is the number of degrees of freedom of the system S , M is the mass of thethermostat and k is Boltzmann’s constant. Nos´e’s thermostated Hamiltonian F hastwo desirable properties: the orbit average of T = (cid:12)(cid:12) ps − (cid:12)(cid:12) is T and the thermostatedsystem is Hamiltonian. A drawback of the Nos´e thermostat is the measure d µ N =exp( − βF ) d p d q d p s d s is not normalizable (i.e. there is no partition function for F ), so phase space averages with respect to the extended phase space variables( q, p, s, p s ) are undefined.Hoover [3] introduced a non-symplectic reduction of Nos´e’s thermostat by elim-inating the state variable s and rescaling time t : q = q, ρ = ps − , dd τ = s dd t , ξ = d s d τ . This reduction has the desirable properties: when E = H ( q, ρ ) + M ξ , themeasure d µ E = exp( − βE ) d q d ρ d ξ is finite and so has a partition function; itprojects to d µ (eq. 1); it is stationary for the reduced thermostat; and when thesystem is a simple harmonic oscillator, the equilibrium statistical mechanical modelpredicts the variates q, ρ and ξ are Gaussian.Indeed, the Nos´e-Hoover thermostated simple harmonic oscillator reduces to thefollowing “simple” system:˙ q = ρ, ˙ ρ = − q − ξρ, ˙ ξ = (cid:0) ρ − T (cid:1) /M. (4)Legoll, Luskin and Moeckel show in [4] that near the decoupled limit of M = ∞ and ξ = 0, the thermostated harmonic oscillator (eq. 4) is non-ergodic. Bymeans of an averaging argument, they reduce the thermostated equations to a non-degenerate twist map to show the existence of KAM tori. The result is generalizedin a subsequent paper to 1-degree of freedom thermostats for which an associatedpotential function G (eq. 33 of [5]) is not isochronous.1.1. The high-temperature limit.
The present paper examines the dynamicsof Nos´e’s thermostat near the high-temperature limit T = ∞ with the thermostatmass M held constant. It presents a proof of the existence of KAM tori based onthe integrability of suitably rescaled equations at the T = ∞ limit. Specifically, Theorem 1.1.
Let V : R / π Z −→ R be C r , r > , and let H : T ∗ R / π Z −→ R be H ( q, p ) = 12 p + V ( q ) . (5) Fix the thermostat mass
M > . The Nos´e-thermostated Hamiltonian F (eq. 3)associated to H possesses invariant KAM tori for all T > sufficiently large. The intuition behind this theorem is the following: for large temperatures, be-cause the potential V is bounded, most of the energy must be kinetic. Therefore,the dynamics should look like a perturbation of the purely kinetic hamiltonian(where V ≡ NVARIANT TORI 3
Alternative Thermostats.
A natural question that arises in light of theabove results on the existence of invariant tori is whether there are thermostats likeNos´e’s that do not possess these invariant KAM tori in the large temperature limit.Let’s say that a Nos´e-like thermostat is one which involves momentum rescalingand the thermodynamic equilibrium (where ˙ s = 0 = ˙ p s ) is independent of thatrescaling. This paper proves that Theorem 1.2.
Let ( N, u ) = ( N T ( s, p s ) , u ( s )) be a thermostat that satisfies (1) N is homogeneous quadratic and increasing in p s ; (2) u : R + −→ R + is an increasing diffeomorphism; (3) for all Hamiltonians H = H ( q, p ) , if F = H ( q, p/u ( s )) + N T ( s, p s ) has athermodynamic equilibrium then it is independent of s .Then, up to a rescaling and change of variables, u = s and there is a smooth positivefunction Ω T = Ω T ( u ) such that N = 12 Ω T p u + nkT ln u. (6) In addition, if Ω T ( u/ √ T ) T −→∞ −→ Ω( u ) in C r ( R + , R + ) for some r > , then theNos´e-thermostated Hamiltonian F associated to H (eq. 5) possesses invariant KAMtori for all T > sufficiently large. This theorem is proven in a manner similar to that of Theorem 1.1. Indeed,Theorem 1.1 can be viewed as a special case of 1.2.1.3.
A Hamiltonian Proof of Non-Ergodicity of the Thermostated Har-monic Oscillator.
It is common in the analysis of the Nos´e-Hoover thermostatto fix the temperature T = 1 and let the thermostat mass M −→ ∞ (the weak-coupling limit). This is not equivalent to fixing the thermostat mass M = 1 andletting T −→ ∞ (the high-temperature limit), see (eq. 8) below, but the methodused in the proof of Theorem 1.1, along with first-order averaging, yields a proofof the following theorem, first proven in [4]. Theorem 1.3.
Let ω > and H ( q, p ) = 12 p + 12 ( ωq ) . (7) Fix the temperature
T > . The Nos´e-thermostated Hamiltonian F (eq. 3) associ-ated to H possesses KAM tori for all ǫ = 1 / √ M > sufficiently small. Terminology and Notation
Generating functions provide a convenient way to create canonical transforma-tions. To explain, let ( q ′ , p ′ ) = f ( q, p ) be a canonical transformation, so that q ′ · dp ′ + p · dq = dϕ is closed and therefore locally exact. That is, there is a locally-defined function ϕ = ϕ ( p ′ ; q ) of the mixed coordinates ( p ′ ; q ) such that q ′ = ∂ϕ/∂p ′ and p = ∂ϕ/∂q . The transformation f is implicitly determined by ϕ . The identitytransformation has the generating function ϕ = q · p ′ .In the sequel, a canonical system of coordinates ( x, X ) = ( x , . . . , x n , X , . . . , X n )are denoted using the capitalization convention: the Liouville 1-form equals P ni =1 X i d x i and X i is the momentum conjugate to the coordinate x i .The KAM theorem gives sufficient conditions which imply that a sufficientlysmooth perturbation (say C r for r > n ) of an integrable n -degree of freedom LEO T. BUTLER
Hamiltonian has invariant tori. A Hamiltonian which satisfies one of these sufficientconditions is said to be KAM sufficient.In practice, construction of action-angle coordinates for a particular Hamiltonianis a very difficult problem. However, approximate action-angle coordinates may beconstructed by methods similar to their construction in the Birkhoff Normal Form:by means of a sequence of generating functions that transform the Hamiltonian intoa near-integrable form. In this case, one verifies KAM sufficiency for the integrableapproximation. 3.
The Rescaled Thermostat
Let us rescale the variables in the Nos´e thermostat so that the Boltzmann con-stant k = 1 and q = √ M w mod 2 π, p = W/ √ M , s = σ/ √ M T , p s = √ M T Σ . (8)With this canonical change of variables, the thermostated Hamiltonian for H (eq.5) is ( ǫ = 1 / √ M ) F = T × (cid:20)
12 (
W/σ ) + 12 Σ + βV ( w/ǫ ) + ln σ (cid:21)| {z } F β − T ln( M T ) . (9)Since the coordinates ( w, σ ) and ( W, Σ) are canonically conjugate, up to a rescalingof time by the factor T , the Hamiltonian flow of F equals that of F β .4. KAM tori in the high-temperature limit
Because the timescale of the thermostat, ǫ , enters into the rescaled thermostatedHamiltonian F β only through the bounded potential V , and the analysis of thissection focuses on the high-temperature limit β −→ + , the convention is adoptedthat M = 1 ( = ⇒ ǫ = 1) . (10)The analysis below is altered in insignificant ways by this additional hypothesis. Lemma 4.1.
Let β = 0 . Under the canonical change of coordinates induced byintroducing cartesian coordinates, ( a, b ) = ( σ cos w, σ sin w ) , (11) the rescaled thermostated Hamiltonian equals F = 12 (cid:2) A + B (cid:3) + 12 ln (cid:0) a + b (cid:1) . (12) That is, F is a mechanical hamiltonian with a rotationally invariant potential. The proof is a simple computation. With the interpretation that F is theHamiltonian of the thermostated free particle ( V ≡ F along the varietyΞ = { ( σ, w, Σ , W ) | σ = | W | 6 = 0 , Σ = 0 } , (13)with each periodic orbit parameterized by the angular momentum integral µ = W .Ideally, one would like to apply a theorem of R¨ussmann and Sevryuk [8, 9]. In this NVARIANT TORI 5 context the theorem says that if the ratio of periods T /T of the periodic orbit andthe linearized reduced hamiltonian is not constant, then F is KAM-sufficient, i.e.invariant KAM tori survive for F β for all β sufficiently small. Unfortunately, thepotential functions U ( σ ) = σ α /α (including the degeneration, U = ln, at α = 0)are characterized by constancy of this ratio.Instead, we compute an approximate change of coordinates to action-angle vari-ables using a succession of generating functions.As noted above, F has an invariant family of periodic orbits along the varietyΞ, with each periodic orbit Ξ µ = { ( | µ | , w, , µ ) | w ∈ R / π Z } parameterized byangular momentum µ = 0. On the other hand, let T ∗ T have the canonical coordi-nates { ( θ, η, I, J ) | θ, η ∈ R / π Z , I, J ∈ R } and let Z ⊂ T ∗ T be the zero section { ( θ, η, , } . Lemma 4.2.
There are open sets A ⊂ T ∗ T , B ⊂ T ∗ ( R + × T ) such that Z ⊂ A , Ξ ⊂ B and a canonical transformation Φ : A − Z −→ B − Ξ ( σ, w, Σ , W ) = Φ( θ, η, I, J ) that transforms the Hamiltonian F (eq. 9) to F = I ( − I + 1 + J + J ) − J (1 + J/ J / J /
4) + O (5) (14) where I has degree , J has degree and O (5) is a remainder term containing termsof degree ≥ .Remark . The transformation Φ extends continuously over the zero section Z .The extension blows down the 2-torus Z to the 1-torus (periodic orbit) Ξ bycollapsing the θ -cycle on Z . In addition, the non-standard choice of degrees forthe action variables I and J is because they are determined by the pullback of thedegrees of σ, w, Σ and W (all of degree 1) by Φ. Proof.
The generating function ϕ ( W, Σ; u, v ) = (1 − u ) W Σ + (1 − W ) v induces thecanonical transformation ( σ, w, Σ , W ) = f ( u, v, U, V ) where σ = (1 − u )(1 − V ) , w = − v − U (1 − u ) / (1 − V ) mod 2 π, (15)Σ = U/ ( V − , W = 1 − V. This transforms the Hamiltonian F to F = 12 (1 − u ) − + 12 (1 − V ) − U + ln(1 − u ) | {z } G + ln(1 − V ) . (16)The symplectic map f is singular along the set { V = 1 } (which should be mappedto the zero angular momentum locus { W = 0 } ), and it transforms { u = 0 , U = 0 } to the variety of periodic points Ξ. By design, f is a symplectomorphism of { ( u, v, U, V ) | V < } ⊂ T ∗ ( R × R / π Z ) onto an open neighbourhood of Ξ. Addi-tionally, f maps an open neighbourhood of { ( u, v, U, V ) | u = U = V = 0 } onto anopen neighbourhood of the periodic locus Ξ .The determination of a further coordinate change is independent of the finalterm in F , which involves only V , so let G = F − ln(1 − V ) as indicated in (eq.16). With the fourth-order Maclaurin expansion of G , one obtains G = (cid:18) V V + 12 (cid:19) U + (cid:18) u u (cid:19) u + O (5) , (17) LEO T. BUTLER where O (5) is the remainder term that contains terms of degree 5 and higher.One postulates a second generating function ν = ν ( U, V ; x, y ) = xU + yV + X ≤ i + j + k + l ≤ ν ijkl x i y j U k V l + O (5) , (18)and a transformed Hamiltonian G = (cid:18) x + X (cid:19) (cid:18) α (cid:18) x + X (cid:19) + γY + βY + 1 (cid:19) + O (5) . (19)One solves for the generating function ν and G simultaneously, and arrives at ν = yV + Ux − UV x − Ux + UV x + UV x + U x (20)+ xU − U V − U + O (5)and α = − / β = γ = 1.Finally, let I = ( x + X / θ be the conjugate angle (mod2 π ), and η = y mod2 π , J = Y . Then the transformed Hamiltonian F is congruent mod O (5) to thatin (eq. 14). (cid:3) Proof of Theorem 1.1.
The rescaled thermostated Hamiltonian F β = F + βV ( q ) = F + O ( β ) where O ( β ) = βV ( w ) is C r , r >
4, and 2 π -periodic in w . Underthe sequence of canonical transformations in lemma 4.2, w = − η + ρ ( θ, η, I, J ) + O (5) mod 2 π where ρ is an analytic real-valued function, and O (5) is a remainderin I, J . So the perturbation in the approximate angle-action variables ( θ, η, I, J ) is C r , r >
4, and O ( β ).Since F (eq. 14) has a non-vanishing Hessian determinant in the action variables( I, J ), the KAM theorem applies [10, 2, 1, 6]. (cid:3) Nos´e-like Thermostats
This section proves theorem 1.2. This section employs the convention that G i denotes the partial derivative of the function G with respect to the i -th variable.5.1. The Thermostat’s Normal Form.
To prove the normal form for a Nos´e-like thermostat in 1.2, observe that Hamilton’s equations for the Hamiltonian F ( q, p, s, p s ) = H ( q, p/u ) + N ( s, p s ) are˙ q = u − H , ˙ p = − H , (21)˙ s = N , ˙ p s = u ′ u E ( H ) − N , where H i ( N i ) is the partial derivative of H ( N ) with respect to the i -th argument, E ( H ) ( q,p ) = p · H ( q, p ) is the fibre derivative of H and H and its derivatives areevaluated at ( q, p/u ).In thermodynamic equilibrium, ˙ s = 0 = ˙ p s . Solving ˙ p s = 0 yields E ( H ) = N / (ln u ) s . Since the right-hand side is independent of ( q, p ), the left-hand sidemust be depend only on s and therefore it must be constant. Following convention, A reader who is familiar with the Birkhoff normal form may wonder why G includes cubicterms. These computations mirror those for the Birkhoff normal form, but our Hamiltonian is notbeing expanded in a neighbourhood of an isolated critical point. NVARIANT TORI 7 let nkT be this constant. Then, since ˙ s = 0 and N is increasing and homogeneousof degree 2 in p s , N ( s, p s ) = 12 Ap s + nkT ln u, (22)where A = A ( s ) >
0. Because T is constant, the function A may be parameterizedby T so: A = A T . Since u is a diffeomorphism, the change of variables s −→ u gives N ( u, p u ) = 12 Ω T p u + nkT ln u, (23)where Ω T = A T · ( u ′ ) . This proves the normal form for the thermostat under thehypotheses of 1.2. Remark . In the general case where N vanishes along p s = 0, A = A ( s, p s ) is asmooth function of both variables. This added generality introduces the possibilityof multiple thermodynamic equilibria at the same temperature, which differ onlyin the value of the momentum p s . It is difficult to understand the significance ofthis.5.2. KAM-tori in the high-temperature limit.
By means of the rescaling ineq. 8, with M = 1, the thermostated Hamiltonian is transformed to F = T × (cid:20)
12 (
W/σ ) + 12 Ω T ( σ/ √ T )Σ + βV ( w/ǫ ) + ln σ (cid:21)| {z } F β − T ln( T ) . (24)By the hypothesis of Theorem 1.2, as T −→ ∞ , Ω T ( σ/ √ T ) converges in C r ( R + , R + )to a limit Ω( σ ) for some r > F has the invariant variety Ξ (eq. 13) of periodic points andthe invariant periodic set Ξ , as in the constant thermostat mass case. Lemma 5.1.
Assume that Ω( σ ) = 1 + a ( σ −
1) + b ( σ − / · · · . If b =(96 α + 9 a − a + 44) / , β = (2 − a ) / and γ = (48 α + 3 a − a + 34) / , thenthere are open sets A ⊂ T ∗ T , B ⊂ T ∗ ( R + × T ) such that Z ⊂ A , Ξ ⊂ B and acanonical transformation Φ : A − Z −→ B − Ξ ( σ, w, Σ , W ) = Φ( θ, η, I, J ) that transforms the Hamiltonian F (eq. 24) to F = I ( αI + 1 + βJ + γJ ) − J (1 + J/ J / J /
4) + O (5) (25) where I has degree , J has degree and O (5) is a remainder term containing termsof degree ≥ .Remark . In the case a = b = 0, one finds that α = − /
24 and β = 1 = γ ,which is the result of Lemma 4.2. Similar to the assumption that M = 1 in theNos´e-thermostat case, the assumption that the inverse mass Ω(1) = 1 simplifiesthe statement of Lemma 5.1 and the proof of Theorem 1.2, but the latter Theoremholds for any value of Ω(1) > a and b of the thermostat’s inverse mass Ω and α, β, γ of the normal form in approximate action-angle variables arise from theattempt to force the normal form to be I . LEO T. BUTLER
Proof of Theorem 1.2.
By Lemma 5.1, the determinant of the Hessian of F withrespect to the action variables I, J is − (cid:0) α + β (cid:1) + 4 αγI − βγ + α ) J − (cid:0) γ + 6 α (cid:1) J + O (3) , which equals O (3) iff α = β = γ = 0. However, if α = 0 = γ , then a = 2 and so β = 0. (cid:3) The Harmonic Oscillator in the Weak-Coupling Limit
Proof of Theorem 1.3.
After applying the change of variables in eq. 8, and a rescal-ing of (
W, w ) the rescaled thermostated harmonic oscillator Hamiltonian is G κ = 12 ( W/σ ) + 12 w + κ (cid:18)
12 Σ + ln σ (cid:19) , (26)where κ = ǫ/ ( ω √ β ). In the following, it will be assumed that ω = β = 1 so that κ = ǫ . The generating function ϕ = wV √ σ + σU induces the canonical change ofvariables σ = u, w = v/ √ u, Σ = U + 12 vV /u, W = V √ u. (27)When composed with the canonical transformation u −→ − u , U −→ − U , theHamiltonian G κ transforms to G κ = 12(1 − u ) (cid:0) V + v (cid:1)| {z } E + κ (cid:20)
12 ( U − vV / (1 − u )) + ln(1 − u ) (cid:21)| {z } Q κ . (28)This Hamiltonian weakly couples the variables ( v, V ) with ( u, U ) when κ << v, V ) evolving on a fast time-scale and ( u, U ) evolving on a slow timescale.Averaging the Hamiltonian G κ in ( v, V ) over a period gives¯ G κ = E − u ) + κ (cid:20) U + E (1 − u ) + ln(1 − u ) (cid:21) + O ( κ ) (29)The Hamiltonian κ − ¯ G κ has a second-order Maclaurin expansion of12 U + (cid:18) E
16 + Eκ − (cid:19) u + (cid:18) E Eκ − (cid:19) u + E
16 + Eκ + O ( κ ) . (30)When E = κ + O ( κ ), κ − ¯ G κ has a critical point at u = U = 0 and the fourth-orderMaclaurin expansion is12 (cid:0) U + u (cid:1) + 2 u u O ( κ ) (31)Computations similar to those in Lemma 4.2 show that the Birkhoff Normal Formis ¯ G κ = κI (1 − I/
24) + O ( κ , , (32)where I = (cid:0) U + u (cid:1) . Since the averaged system is KAM sufficient, the unaver-aged Hamiltonian G κ is an O ( κ ) perturbation of a KAM sufficient Hamiltoniansystem. (cid:3) NVARIANT TORI 9
Remark . One may attempt to apply the Birkhoff Normal Form to the Hamil-tonian ˆ G κ = G κ − κu/ (1 − u ) . The generating function ν = U x − U x / U x /
288 + 295 U x/ − U / V y + U ( x − y + V ) / κ + U V y/ κ induces a canonical transformation ( u, v, U, V ) = f ( θ, η, I, J ) that transforms ˆ G κ to normal form: ˆ G κ = κI + J + αI + βIJ + γJ + O (5) , (34)where α = − κ/ β = − γ = − / κ and I = ( x + X ) / J = ( y + Y ) / J = 0, ˆ G κ in (eq. 34) coincides with the averaged Hamiltonian ¯ G κ in (eq. 32). If the total energy is fixed at ˆ G κ = κh , then J = κ (cid:0) h + h / − I + I /
24 + O (5) (cid:1) (35)is the Hamiltonian of the reduced system dθ/dη = − ∂J/∂I , dI/dη = 0 on theisoenergy level ˆ G κ = κh . This implies that the Hamiltonian ˆ G κ is KAM sufficient[10, pp. 46–47].The final step in this line of proof would be to prove that in the limit at κ = 0 of asuitably renormalized ˆ G κ is KAM sufficient and G κ is a suitably small perturbation.7. Conclusion
This note has demonstrated the existence of KAM tori near the high-temperaturelimit of a Nos´e-thermostated 1-degree of freedom system with a periodic potential,along with similar thermostats that are scale-invariant. It has also given a “Hamil-tonian” proof of Legoll, Luskin and Moeckel’s result on the existence of KAM toriin the Nos´e-Hoover thermostated harmonic oscillator in the weak-coupling limit.It is expected that the techniques of this paper may be used to demonstratesimilar results for n -degree of freedom Nos´e-thermostated systems. Potentiallymore fruitful, however, is that the techniques of this paper might be useful tocreate thermostats with the desired properties. Of course, some features of theNos´e-type thermostat must be abandoned in the process. References
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Department of Mathematics, North Dakota State University, Fargo, ND, USA, 58108
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