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Phase-Space Metric for Non-Hamiltonian Systems

We consider an invariant skew-symmetric phase-space metric for non-Hamiltonian systems. We say that the metric is an invariant if the metric tensor field is an integral of motion. We derive the time-dependent skew-symmetric phase-space metric that satisfies the Jacobi identity. The example of non-Hamiltonian systems with linear friction term is considered.


Introduction
The dynamics of Hamiltonian systems is characterized by conservation of phase-space volume under time evolution. This conservation of the phase volume is a cornerstone of conventional statistical mechanics of Hamiltonian systems. At a mathematical level, conservation of phasespace volume is considered as a consequence of the existence of an invariant symplectic form (skew-symmetric phase-space metric) in the phase-space of Hamiltonian systems [1,2,3].
Tuckerman et al. have argued [5,6] that there is a measure conservation law that involves a nontrivial phase-space metric. This suggests that phase-space should be carefully treated using the general rules of the geometry of manifolds [1,2]. Tuckerman et al. have applied the concepts of Riemannian geometry to the classical statistical mechanics of non-Hamiltonian systems [4,5,6]. Tuckerman et al. have argued that, through introduction of metric determinant factors g(x, t), it is possible to define an invariant phase-space measure for non-Hamiltonian systems.
In their approach the metric determinant factor g(x, t), where g(x, t) is the determinant of the metric tensor, is defined by the compressibility of non-Hamiltonian systems. However Tuckerman et al. consider only the determinant g(x, t) of the metric. The phase-space metric is not considered in [4,5,6]. Note that Tuckerman et al. suppose that the metric determinant factor is connected with symmetric phase-space metric. It can be proved that the proposal to use an invariant time-dependent metric determinant factor in the volume element corresponds precisely to finding a skew-symmetric phase-space metric (symplectic form) that is an integral of motion. Therefore we must consider the skew-symmetric phase-space metric.
Sergi [10,11] has considered an antisymmetric phase-space tensor field, whose elements are general function of phase-space coordinates. In [10,11], the generalization of Poisson brackets for the non-Hamiltonian systems was suggested. However the Jacobi identity is not satisfied by the generalized brackets and skew-symmetric phase-space metric. As a result the algebra of phase-space functions is not time translation invariant. The generalized brackets do not define a Lie algebra in phase-space. Note that the generalized brackets of two constant of motion is no longer a constant of motion.
In the present paper we consider an invariant skew-symmetric (antisymmetric) phase-space metric for non-Hamiltonian systems. We say that the metric is an invariant, if the metric tensor field is an integral of motion. We define the phase-space metric such that the Jacobi identity is satisfied. The suggested skew-symmetric phase-space metric allows us to introduce the generalization of the Poisson brackets for non-Hamiltonian systems such that the Jacobi identity is satisfied by the generalized Poisson brackets. As a result the algebra of phase-space functions is time translation invariant. The generalized Poisson brackets define a Lie algebra in phase-space. The suggested Poisson brackets of two constant of motion is a constant of motion.
In section 2, the definitions of the antisymmetric phase-space metric, mathematical background and notations are considered. In section 3, we define the non-Hamiltonian systems, and consider the Helmholtz conditions. In section 4, we consider the time evolution of phase-space metric. We derive the phase-space metric that is an integral of motion. In section 5, the generalized Poisson brackets for non-Hamiltonian systems are defined. In section 6, the example of phase-space metric for non-Hamiltonian system with the linear friction term is considered. Finally, a short conclusion is given in section 7.

Phase-space metric
The 2n-dimensional differentiable manifold is denoted M. Coordinates are x = (x 1 , ..., x 2n ). We assume the existence of a time-dependent metric tensor field ω kl (x, t) on the manifold M. We can define a differential 2-form where ω kl = ω kl (x, t) is a skew-symmetric tensor ω kl = −ω lk , and the tensor elements ω kl (x, t) are explicit functions of time. Here and later we mean the sum on the repeated index k and l from 1 to 2n. We suppose that the differential 2-form ω is a closed nondegenerated form: 1) If the metric determinant is not equal to zero for all points x ∈ M, then the form ω is nondegenerated.
Phase-space is therefore assumed to be a symplectic manifold.

Definition 1.
A symplectic manifold is a differentiable manifold M with a closed nondegenerated differential 2-form ω.
For symplectic manifold, we have the phase-space volume element This differentiable 2n-form can be represented by where g(x, t) is defined by equation (2). The nondegenerated condition (g(x, t) = 0) for the metric ω kl (x, t) is equivalent to the condition ω n = 0 or v = 0. It is known [2] that there exists the local coordinates (q, p) such that Here and later we mean the sum on the repeated index i and j from 1 to n.

Non-Hamiltonian system
The dynamics is described by a smooth vector field X = X(x), with components X k in basis ∂ k = ∂/∂x k . For simplicity, we consider the case where the vector field X is time independent. In local coordinates {x k }, equation (5) has the form Consider now the definition of the Hamiltonian systems, which is used in [1].
where ω(X) = i X ω is the contraction (interior product) of the 2-form ω with vector X, and d is the exterior derivative.

Proposition 1. The classical system (5) is a Hamiltonian system if the conditions
are satisfied.
Proof. In the local coordinates {x k }, we have where X k = ω kl X l . In this case, the exterior derivative of 1-form ω(X) is As the result we have the differential 2-form This differential 2-form is a symplectic form, which can be called "non-Hamiltonian symplectic form". If the Helmholtz conditions (7) are satisfied, then the differential 1-form ω(X) is closed (dω(X) = 0), and the classical system (5) is a Hamiltonian system.
Proof. In the canonical coordinates (q, p), the vector field X has the components (G i , F i ), which are used in equation (9). The 1-form ω(X) is defined by the following equation: The exterior derivative for this form can now be written by the relation It now follows that This equation can be rewritten in an equivalent form Here we use the skew-symmetry of dq i ∧ dq j and dp i ∧ dp j with respect index i and j. It is obvious that conditions (10) - (12) lead to the equation dω(X) = 0.

Time evolution of phase-space metric
Let us find a time-dependent symplectic 2-form ω that satisfies the equation dω/dt = 0. It is known [2,3] as the following proposition.
Proof. The proof of this theorem is considered in [2,3].
Let us consider a generalization of this proposition. Proposition 3. If the time-dependent metric ω kl = ω kl (x, t) is a skew-symmetric metric (ω kl = −ω lk ) that is satisfied by the Jacobi identity (3), and the system is defined by equation (6), then the total time derivative of the differential 2-form (1) is given by Proof. The time-derivative of the time-dependent symplectic form ω is given by Then, using the equation and equation (6), we find that This expression can be rewritten in an equivalent form Using the rule of term-by-term differentiation in the form we get the following equation: Using the Jacobi identity (3), and skew symmetry ω ml = −ω lm , ω km = −ω mk , we have As the result, we obtain equation (13) for the total time derivative of symplectic form.
The total time derivative of the symplectic form is defined by equation (13). If the total derivative is zero, than we have the integral of motion or invariant. It is easy to see that the differentiable 2-form ω is invariant if the phase-space metric ω kl (x, t) is satisfied by the equation This equation can be rewritten in an equivalent form where the operatorĴ is defined by the equation Proposition 4. The differentiable 2-form ω is invariant (is an integral of motion for non-Hamiltonian system (6)) if the phase-space metric ω kl (x, t) is defined by the equation HereĴ is an operator that is defined by equation (15).
Proof. Let us consider the formal solution of equation (14) in the form where ω lk . In this case, the time-independent tensor fields ω (n) kl (x) are defined by the recursion relation ω km X m ). This equation can be rewritten in an equivalent form Using the skew symmetry of the ω (n) ns , we have This relation can be represented in the form where the operatorĴ is defined by equation (15). Therefore the invariant phase-space metric is defined by the following equation: As the result we have equation (16).
Comments. We prove that equation (14) for the elements of phase-space metric can be expressed in terms of an operatorĴ such that whereĴ ab kl is defined by equation (15). In this case, the time evolution of the phase-space metric from initial condition ω kl (x 0 , 0) to a value ω kl (x, t) at the time t can be written by the equation Here the matrix exponential operator, exp(tĴ ), can be called the metric propagator. The operatorĴ can be considered as a metric analog of the Liouville operator. Note that we can use the canonical coordinates (q, p) for t = 0 and the coordinate-independent initial metric: ω kl (x, 0) = ω It is natural to approximate the short-time propagator exp(tĴ ) using the Trotter theorem [23,24,25]. The vector field X = X(x) can be represented in the form where X 1 is a Hamiltonian term such that and X 2 is a friction (non-Hamiltonian) term. As the result we have An important consequence would be the ability to formulate rigorous numerical integration algorithms based on Trotter-type splittings [23,24,25] of the classical propagator exp(∆tĴ ). The metric propagator for the small time step is Finally, we obtain

Poisson brackets for non-Hamiltonian systems
Let us consider the skew-symmetric tensor field ω kl = ω kl (x, t) that is defined by the equations As the result this tensor field satisfies the Jacoby identity ω kl ∂ l ω ms + ω ml ∂ l ω sk + ω sl ∂ l ω km = 0.
It follows from the Jacoby identity for ω kl .
In proposition 3, we suggest the time-dependent phase-space metric ω kl (x, t), which satisfies the Jacoby identity. As the result we have Lie algebra that is defined by the following brackets: It is easy to prove that these brackets are Poisson brackets.
In the general case, the rule of term-by-term differentiation with respect to time that has the form d dt whereȦ = dA/dt, is not valid for non-Hamiltonian systems. In the general case, we have where Note that time evolution of the Poisson brackets (20) for non-Hamiltonian systems can be considered as t-deformation [28] of the Lie algebra in phase-space.
If we use the invariant phase-space metric, then the rule (21) is valid. As the result the suggested Poisson brackets (20) of two constants of motion is a constant of motion and rule (21) is satisfied.

Example: System with linear friction
Let us consider the non-Hamiltonian system that is described by the equations: where H = T (p) + U(q) and i; j = 1, ..., n. Here T (p) is a kinetic energy, U(q) is a potential energy. The term −K i j (t)p j is a friction term. Usually, this system is described by the phasespace metric ω kl that has the form where the matrix G = ||g ij || is equal to indentity matrix E = ||δ ij ||. Here G T is transpose matrix for the matrix G. The symplectic form is defined by equation (4) in the form ω = 2g ij dq i ∧ dp j = δ ij dq i ∧ dp j , where g ij = (1/2)δ ij . The phase-space compressibility of the system (22) is defined by the spur of the matrix K = ||K i j || in the form Proposition 5. The invariant phase-space metric for the classical system (22) has the form where the matrix G is defined by Proof. Suppose that phase-space metric depends on time t. Therefore the matrix G and elements g kl are the functions of the variable t: G = G(t), g ij = g ij (t). Let us consider the total time derivative of the symplectic form ω = 2g ij (t)dq i ∧ dp j .
As the result, we have In order to have the invariant phase-space metric (dω/dt = 0), we use the following equation: For example, the system has the invariant phase-space metric ω kl (t) in the form In this case, the metric determinant factor is equal to

Conclusion
Tuckerman et al. [4,5,6] suggest a formulation of non-Hamiltonian statistical mechanics which uses the invariant phase-space measure. The invariant measure is connected with phase-space metric. Tuckerman et al. consider the properties of only the determinant g(x, t) of the metric. The phase-space metric is not considered in [4,5,6]. We consider the invariant phase-space metric for non-Hamiltonian systems. The proposal to use an invariant time-dependent metric determinant factor in the volume element corresponds precisely to finding a skew-symmetric phase-space metric (symplectic form) that is an integral of motion. Therefore we consider the skew-symmetric metric. Sergi [10,11] uses the skew-symmetric phase-space metric that is not satisfied by the Jacoby identity. As the result the generalization of the Poisson brackets for non-Hamiltonian systems leads one to non-Lie algebra. Note that non-Lie algebra for non-Hamiltonian systems is considered in [29]. In this paper we consider an invariant antisymmetric phase-space metric that satisfies the Jacoby identity, and defines the Lie algebra in phase-space. We call the metric is invariant if the metric tensor field is an integral of motion (dω/dt = 0). This invariant phase-space metric ω kl (x, t) defines the invariant phase-space measure v by the equation v = (1/n!)ω n = g(x, t)dx 1 ∧ ... ∧ dx 2n , where g(x, t) is the metric determinant g(x, t) = det(ω kl (x, t)) and dv/dt = 0. The suggested time-dependent skew-symmetric phasespace metric leads to a constant value of the entropy density, so that the associated distribution function obeys an evolution equation associated with incompressible dynamical flow.
Note that the invariant phase-space metric of some non-Hamiltonian systems can leads us to the lack of smoothness of the metric. In this case, the phase-space probability distribution can be collapsed onto a fractal set of dimensionality lower than in the Hamiltonian case [26,27]. Unfortunately the description of lack of smoothness in [26,27] is considered without using the curved phase-space approach [4,5,6]. Note that classical systems that are Hamiltonian systems in the usual phase-space are non-Hamiltonian systems in the fractional phase-space [13,32].
In the papers [30,31], the quantization of the evolution equations for non-Hamiltonian and dissipative systems was suggested. Using this quantization it is easy to derive the quantum analog of the invariant Poisson brackets, which satisfy the rule of term-by-term differentiation with respect to time.