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Proof of Irreversibility for Systems with Time Reversal Symmetry

The positivity of entropy change rate was rigorously proven in a certain microscopic system associated with a Hamiltonian using only information about the microscopic system. This microscopic system obeys an equation with time-reversal symmetry. We proved that a symplectic map with time-reversal symmetry has the same ergodic properties as an Anosov diffeomorphism. This result guarantees that any initial density function defined except for a zero volume set converges to the unique invariant density (uniform distribution) in the sense of mixing. Moreover, we proved that the Kolmogorov--Sinai entropy, which is the entropy change rate, is \textit{positive} using the property of Anosov diffeomorphisms. In addition, we discovered a mathematical structure connecting the convergence of initial density functions to the uniform distribution and positivity of the entropy change rate (Anosov diffeomorphism).


Introduction
The problem of time's arrow is considered controversial because it claims that unidirectional time evolution occurs on a macroscopic scale although a microscopic system obeys an equation with time-reversal symmetry. In discussions about time irreversibility, it is important to specify time scales and which quantity evolves irreversibly. In the 19th century, Boltzmann clarified the irreversible behavior of an H-function by assuming the molecular chaos hypothesis [1,2]. Based on Boltzmann's H-theorem, Loschmidt [3] claimed that in a system with time-reversal symmetry, the time derivative of the entropy would be negative in a backward motion. In addition, Zermelo [4] asserted that the value of the entropy would return to the initial value by quoting Poincaré's recurrence theorem. Kac showed how this irreversible behavior occurs in the Ehrenfest model when the initial condition is far from the equilibrium state using the mean recurrence time [5,6]. In the Ehrenfest model, the state of the system evolves stochastically. In the 1970s, for nonequilibrium systems in stationary and chaotic states, the idea of the SRB distribution was proposed by Sinai, Ruelle, and Bowen through mixing Anosov diffeomorphisms, in which it was proved that there is a unique ergodic distribution [2,7,8]. If one proves that a dynamical system has an SRB measure, then it is guaranteed that initial density functions converge to the SRB density function (unique ergodic density function) [8,9] in the sense of mixing. Using the property of the Anosov diffeomorphisms and SRB measure, one obtains the positive Kolmogorov-Sinai (KS) entropy h KS . It is known that for Anosov systems, by coarsegraining, the Boltzmann entropy can be denoted as a linear function in terms of time whose slope is represented by the Kolmogorov-Sinai entropy for certain time ranges by assuming appropriate conditions [10,11,12].
As an example of a physical model having an SRB measure (a mixing invariant density function), Sinai [13,14] proved the hyperbolicity, mixing property, and ergodicity of dynamical systems with scatterers (Sinai billiards) [15]. Bunimovich [14,16] also proved these properties in a stadium (Bunimovich stadium). Another example is the Lorentz gas where a periodic continuation of Sinai billiards' boundary is applied without any walls [17].
The Sinai billiards, Bunimovich stadium and Lorentz gas are popular chaotic and mixing dynamical models that have scatterers or walls. In these models, the properties of the dynamical systems such as mixing are caused by the existence of obstacles, scatterers, and their shapes and positions. Then, how about dynamical systems without obstacles, scatterers, or walls? For symplectic maps, the sawtooth map [11,18], Arnold's cat map [19] and multi-baker's map [20] have been known as mixing symplectic maps without scatterers. However, these three maps do not have time-reversal symmetry in that they are not invariant under an operation R 1 such as (p, q) → (−p, q).
It is also important to distinguish time-reversal symmetry from reversibility. If a map T has reversibility, it only means that the map T has an inverse T −1 . However, in the case of time-reversal symmetry, the form of R −1 1 • T −1 • R 1 needs to be consistent with that of T . Then, we can say that the standard map [11,21,22,23] and tangent map [24] do not have time-reversal symmetry although they are reversible. Table 1 compares systems satisfying the conditions of paradoxes proposed by Loschmidt and Zermelo. The condition of paradox proposed by Loschmidt is that the system has time-reversal symmetry in the sense of an operator R 1 : (p, q) → (−p, q).
In this paper, our maps derived from a Hamiltonian have time-reversal symmetry in the sense of an operator R 1 : (p, q) → (−p, q) without scatterers or walls, which is different from the previous studies in which following maps or systems are used such as maps without time-reversal symmetry or systems with scatterers or walls.
In this study, a "macroscopic unidirection" is defined by the convergence of initial density functions to the uniform distribution and positivity of the entropy rate on the time scale of relaxation. We propose a symplectic map with time-reversal symmetry derived from a Hamiltonian without scatterers or walls. In our model, for certain parameter ranges we proved that any initial density function defined except for a zero volume set relaxes uniquely to the uniform distribution and that the system is mixed using only information about the microscopic dynamical system [8] and the KS entropy T ε,∆τ , T ε,∆τ standard map tangent map multi-baker's map (proposed in this paper) Reversibility is positive. While proving these properties, we showed that the maps have Anosov properties.

Let us introduce our Hamiltonian
where ε ∈ R is a perturbation parameter, p 1 , p 2 ∈ R and q 1 , q 2 ∈ I δ N,∆τ ≡ Figures 1 and 2 show the shape of the potential V (q 1 , q 2 ) when δ N,∆τ is small enough at ε = 2.0, −2.0, respectively. Then we consider the following map T ε,∆τ obtained using the leap frog method (second order time-reversal symmetric symplectic integrator) for the canonical equations of motion associated with (1) .
    p 1 (n + 1) p 2 (n + 1) q 1 (n + 1) q 2 (n + 1) tan π q 1 (n) + ∆τ 2 p 1 (n) − q 2 (n) − ∆τ 2 p 2 (n) mod I δN,∆τ q 2 (n) + p 2 (n)∆τ + ε(∆τ ) 2 2 tan π q 1 (n) + ∆τ where ∆τ represents a step size. The definition of Γ is a set {(p 1 , p 2 , q 1 , q 2 )} in which there exists an integer n such that T n ε,∆τ (p 1 , p 2 , q 1 , It then turns out that the Lebesgue measure of Γ is zero. The definition of " mod I δ N,∆τ " is as follows.  Figure 1: Shape of the periodic potential V (q 1 , q 2 ) for ε = 2.0 (repulsive force occurs). Definition 1 (mod I δ N,∆τ ). We define an operation mod I δ N,∆τ : R → I δ N,∆τ to be In this study, a map T was called to esnure time-reversal symmetry when the form of R 1 • T was the same as that of T −1 • R 1 by changing the sign of the momentum. The mathematical definition is provided in Section 3. That is, if one chooses one of the two time directions as the "forward direction" and when entropy increases in that direction, then the entropy decreases in the "inverse direction". However, for systems with timereversal symmetry, this time evolution in the "inverse direction" occurs as time evolution in the "forward direction" through the operator R 1 . In our study, we distinguished between "reversibility" and "time-reversal symmetry". The following standard map can be illustrated as such an example as it does not have time-reversal symmetry despite being reversible.
The inverse map is such that Then, the standard map does not satisfy the condition of Loschmidt (time-reversal symmetry in the sense of an operation R 1 : (p, q) → (−p, q)).
In the case of tangent map [24], the time evolution is denoted as The inverse map is such that Then, the tangent map does not satisfy the condition of Loschmidt as R 1 • T tangent is not consistent with T −1 tangent • R 1 . In the case of multi-baker's map [20], the inverse map is given by Note that there is an involution R ′ instead of R 1 such that and it satisfies R ′ • B = B −1 • R ′ . However, in multi-baker's map, one cannot determine which variable has a momentum dimension between x and y. That means we cannot determine which variable represents momentum among x and y. Then, we can say that multi-baker's map does not have a pair of momentum and position (p, q) and this map does not express the equation of motion directly. Thus, in this study, multi-baker's map is not considered to have time-reversal symmetry in the sense of Loschmidt (in the sense of changing the sign of the momentum p → −p). Similarly, Arnold's cat map does not have time-reversal symmetry although they are reversible. On the other hand, the map T ε,∆τ has time-reversal symmetry in terms of p 1 → −p 1 , p 2 → −p 2 . The proof of this time-reversal symmetry is provided in Section 3.
The condition of paradox proposed by Zermelo is that the system satisfies the condition of Poincaré's recurrence theorem. That is, the system needs to preserve the invariant measure [25,26]. For example, our map T ε,∆τ , Hamiltonian systems, Arnold's cat map [19], multi-baker's map [19,20], the standard map [11,21,22,23] and tangent map [24] preserve the Lebesgue measure. Therefore, these systems satisfy the condition proposed by Zeromelo. Therefore, our map T ε,∆τ satisfies the conditions of paradoxes proposed by both Loschmidt and Zermelo.
To prove the convergence of the initial density functions to the uniform distribution and positivity of the entropy rate, we have to show that the dynamical system has Anosov properties [8]. To this end, we constructed other forms of the symplectic map denoted as T ε,∆τ and T ε,∆τ derived from T ε,∆τ to shorten the proof. For the Hamiltonian H, maps T ε,∆τ , T ε,∆τ and T ε,∆τ , the following diagram holds: Let us define the map T ε,∆τ . In the map T ε,∆τ , there are two relations such that where q 0 ≡ q 1 (0) + q 2 (0). Then, by eliminating p 2 and q 2 using the above equation, one defines the map T ε,∆τ as follows.
The symplectic map T ε,∆τ is topologically semi-conjugate with T ε,∆τ using a transformation H. For the symplectic maps T ε,∆τ , T ε,∆τ and T ε,∆τ the following diagram holds: The Jacobian of the map T ε,∆τ is given by and it is not a constant matrix. The case of constant matrices was studied by Franks [27]. The linear instability condition (for at least one eigenvalue of the Jacobian, its absolute value is larger than unity) for the map T ε,∆τ is straightforwardly obtained as and this is related to the condition that the map is an Anosov diffeomorphism (Anosov system).

Time reversal symmetry
In this section, it is shown that the symplectic maps, T ε,∆τ and T ε,∆τ , used in this work have time-reversal symmetry.
Definition 4. Let M be either R 2 × I 2 δ N,∆τ \Γ or M ′ and f : M → M be a map. It is said that map f has time-reversal symmetry when there exists operator R 1 such that where R 1 satisfies R 1 (p, q) = (−p, q) [28].
When there is operator R 1 , the form of f −1 •R 1 is equivalent to that of R 1 •f . Therefore, any orbit obtained by {f −n } n∈Z is also obtained by {R 1 • f n • R −1 1 } n∈Z . For any point (p(n), q(n)), it holds that One can also obtain Therefore, from Definition 4, (23), and (24), it holds that map T ε,∆τ has timereversal symmetry.
In terms of convergence of the initial density function to the equilibrium denisty function, when a dynamical system is exact, any initial density function converges to the unique equilibrium function in the sense of L 1 norm [29]. In the Appendix, we show that the exactness is not compatible with time-reversal symmetry.

Compactness and diffeomorphism
In this section, it is shown that the domain M is compact and T ε,∆τ is a diffeomorphism on M. Domain π . Then, domain M is topologically conjugate with torus T 2 ; thus, M is compact. Next, we show that map T ε,∆τ is a diffeomorphism on M. To this end, one must show that map T ε,∆τ is surjective, injective, and smooth and that its inverse is continuous.
(Proof of injective): We show that map T ε,∆τ is injective by contradiction. Let (s n , t n ) and (s ′ n , t ′ n ) be two points that cannot be identified on M. If one assumes that image Based on the fact that 2t + 2ε(∆τ ) 2 tan(πt) is finite for t ∈ I δ N,∆τ and (29), one obtains the relation s n = s ′ n . These results are in contradiction to the condition (s n , t n ) = (s ′ n , t ′ n ). Therefore, map T ε,∆τ is injective. (Proof of smoothness): It is evident that g, h, ∂g ∂t , ∂h ∂s , and ∂h ∂t defined in (19) are π . Let us discuss the smoothness at 1 2 π . It is evident that g(t) = t and ∂g ∂t are continuous and that h(s, t) = 2t + 2ε(∆τ ) 2 tan(πt) − s and ∂h ∂s are continuous at in a similar manner. Let us discuss the continuity of h(s, t) and ∂h

Mixing Anosov Property
The following theorems give the proof of convergence of initial density functions to the uniform distribution and positivity of the entropy rate for certain parameter ranges.
The definition of an Anosov diffeomorphism [2,8,19,26,27] and the proofs of these theorems are given as follows. Although Gallavotti assumed the chaotic hypothesis [2,8] in many-particle systems, we proved directly that the systems with few degrees of freedom have Anosov properties when the condition (21) is satisfied.

Anosov diffeomorphism and mixing property
In this subsection, we prove that map T ε,∆τ defined by (19) is an Anosov diffeomorphism for ε < 0 or 2 π(∆τ ) 2 < ε. A point, (s n , t n ), of M is denoted as x n = (s n , t n ). The Jacobian of map T ε,∆τ is given by where the components of Jacobian J(x n ) are on R. An Anosov diffeomorphism defined according to [2,8,19,26,27] is given below.
Definition 1 (Anosov diffeomorphism). Let M * be a compact manifold, · be a norm, and f : M * → M * be a diffeomorphism. Consider the tangent space, T x M * , at each point x ∈ M * . f is an Anosov diffeomorphism if it satisfies all of the following conditions: There exist K > 0 and 0 < λ < 1 determined by x and not by ξ or n such that The linear instability condition (for at least one eigenvalue of matrix (30), this eigenvalue's absolute value is larger than unity) for map T ε,∆τ is The linear instability condition is related to the condition that map T ε,∆τ is an Anosov diffeomorphism for map T ε,∆τ . We prove that map T ε,∆τ is an Anosov diffeomorphism for ε < 0, 2 π(∆τ ) 2 < ε. This proof is based on the idea used in [11]. The Jacobians for Arnold's cat map and multi-baker's map are constant. However, the Jacobians for our maps are not constant for any point, and the aforementioned maps are extremely ideal compared with our maps. Thus, it is more difficult to prove that our maps are Anosov diffeomorphisms, as compared to Arnold's cat map and multi-baker's map. To prove this, conditions (I), (II), and (III) in Definition 1 should be satisfied. In order to prove that these conditions are satisfied, we first introduce the definitions of cones L + and L − and then show the following Lemmas: Definition 2. For any two-dimensional nonzero vector on T xn M, x n ∈ M, n ∈ Z, a n = (a 1 (n), a 2 (n)) = O, we define cones L + and L − as , a 2 (n)); J(x n )a n > a n }, ε,∆τ x n ) −1 a n > a n }, where · represents a(n) ≡ a 2 1 (n) + a 2 2 (n). Lemma 3. When the condition ε < 0, 2 π(∆τ ) 2 < ε is satisfied, it holds that J(x n )a n ∈ L + ( T ε,∆τ x n ), ∀ a n ∈ L + (x n ), This Lemma corresponds to condition (II).
From a simple calculation, it holds that Lemma 6. For any point x n and for any integer k ∈ Z ≥ n, there exists a deviation vector, a n , such that Proof. We prove this by contradiction. We assume that One can obtain an orbit, {y m }, including y n and y n ′ , where n ′ > k ′ + 1 > n. For all points, there exists a cone, LL − 1 . Hence, there exists a vector, A n ′ ∈ LL − 1 (y n ′ ). Then, one selects A k ′ +1 and A n as Equation (45) is contradictory to (44). Therefore, Lemma 6 holds.
From Lemmas 3, 4, and 6, we define LL − (x n ) and LL + (x n ) as follows to prove that conditions (I), (II), and (III) are satisfied: From Lemma 6, one can obtain that J(x n )a n ∈ LL − (x n+1 ), ∀ a n ∈ LL − (x n ).
Definition 8. We define a subset, LL + (x n ), of L + (x n ) as Based on simple calculation, it holds that J(x n )a n ∈ LL + (x n+1 ), ∀ a n ∈ LL + (x n ).
Proof. One can select any eigenvector spaces E u x and E s x in LL + (x n ) and LL − (x n ), respectively. Hence, combining (46) and (48) gives Combining (47), (49), and (50), one obtains J( Subspaces E u xn and E s xn are linearly independent by definition, from which it holds that Lemma 10. When the condition ε < 0, 2 π(∆τ ) 2 < ε is satisfied, the condition (III) is satisfied.
Let us define the stretching rate σ ′ (x n , a n ) ≡ n is defined such that sin 2 φ ′ n = a 2 1 (n) a 2 1 (n)+a 2 2 (n) and sin φ ′ n cos φ ′ n = a 1 (n)a 2 (n) a 2 1 (n)+a 2 2 (n) , − π < φ ′ ≤ π. Then, if a n ∈ LL − (x n ), then it holds that There are two cases for ε. The minimum value of σ ′ for each case is calculated as follows.
Proof. From Lemma 9 and Lemma 10, map T ε,∆τ on M is an Anosov diffeomorphism.
According to [19], if maps are Anosov diffeomorphisms, then the corresponding dynamical systems are K-systems. Therefore, the corollary provided below holds.

Anosov property and mixing property
In this subsection, we prove that map T ε,∆τ defined by (15) has the same ergodic properties as an Anosov diffeomorphism for ε < 0 or 2 π(∆τ ) 2 < ε. First, we show that map T ε,∆τ satisfies the conditions (I), (II) and (III) in Definition 1 where M ′ is not conpact. A point, (p 1 (n), q 1 (n)), on M ′ , is denoted as X n = (p 1 (n), q 1 (n)). The Jacobian of map T ε,∆τ is given by where θ n ≡ π 2q 1 (n) + ∆τ p 1 (n) − q 0 − C∆τ n + 1 2 . The linear instability condition for map T ε,∆τ is the same as that in (33). Let us introduce a vector on T Xn M ′ , as follows: we define cones l + and l − as . To prove that map T ε,∆τ satisfies the conditions (I), (II) and (III) in Definition 1, we must show the following Lemmas: Lemma 20. When the condition ε < 0, 2 π(∆τ ) 2 < ε is satisfied, there is a subset, ll + (X n ), of l + (X n ), such that Proof. Let us consider a vector, a ′ n , such that a ′ 2 (n) = 0. The condition a ′ n ∈ l + (X n ) is expressed by where A n , B n , and C n are defined as For any n ∈ Z and X n ∈ M ′ , if (33) is satisfied, then it holds that In this case, condition (65) is rewritten as Selecting a ′ n+1 as J(X n )a ′ n+1 = (a 1 (n + 1), a 2 (n + 1)), one obtains that the condition a ′ n+1 ∈ l + ( T ε,∆τ X n ) is expressed as < 0. Selecting a subset, ll + (X n ), of l + (X n ) as and assuming that a ′ 2 (n + 1) = 0 and a ′ n ∈ ll + (X n ), one obtains that This is contradictory to Here, if the condition . If a ′ n ∈ ll + (X n ) ⊂ l + (X n ), then a ′ n+1 ∈ ll + (X n+1 ) ⊂ l + (X n+1 ).

Lemma 22.
For any point X n and for any integer k ∈ Z ≥ n, there exists a deviation vector, a ′ n , such that Proof. We prove this by contradiction. We assume that One can obtain an orbit, {Y m }, including Y n and Y n ′ , where n ′ > k ′ + 1 > n. For all points, there exists a cone, ll − 1 . Thus, there exists a vector, Thus, Equation (82) is contradictory to (81). Therefore, Lemma 22 holds.
From Lemmas 20, 21, and 22, we define ll − (X n ) and ll + (X n ) as follows to prove that conditions (I), (II), and (III) are satisfied: Definition 23. We define a subset, ll − (X n ), of ll − 1 (X n ) as From Lemma 22, one can obtain that Definition 24. We define a subset, ll + (X n ), of l + (X n ) as Based on simple calculation, one obtains J(X n )a ′ n ∈ ll + (X n+1 ), ∀ a ′ n ∈ ll + (X n ).
Proof. One can select any eigenvector spaces E u X and E s X in ll + (X n ) and ll − (X n ), respectively. Hence, combining (83) and (85) gives Combining (84), (86), and (87), one obtains J(X n )E u Xn ⊂ ll + (X n+1 ) and J(X n )E s Xn ⊂ ll − (X n+1 ). Then, one can inductively select E u X n+1 and E s X n+1 by Subspaces E u Xn and E s Xn are linearly independent by definition, from which it holds that Lemma 26. When the condition ε < 0 or 2 π(∆τ ) 2 < ε is satisfied, the condition (III) is satisfied.
Then, the minimal value of σ(X n , a ′ n ) by varying the vector a ′ n is obtained as 1 + C n . With this minimal value and C n > 0, one has Thus from (97) and (101), by choosing one sees that condition (31) is satisfied. Besides, this relation about λ (λ → 1 − 0 as ε → −0) is equivalent to the fact that the Lyapunov exponent converges to zero value as ε → −0.
Proof. From Lemma 25 and Lemma 26, the map T ε,∆τ satisfies the conditions (I), (II) and (III) in Definition 1 where M ′ is not conpact. It is obvious that the map T ε,∆τ is a diffeomorphism on M ′ . The domain of T ε,∆τ is not compact. However, the map T ε,∆τ is topologically semi-conjugate with T ε,∆τ through the transformation H. That is for any orbit {(p 1 (n), q 1 (n))}, there exists at least one corresponding orbit {(s n , t n )}. Therefore, the dynamical system (M ′ , T ε,∆τ , µ) has the same ergodic properties as (M, T ε,∆τ , µ).
Similar to the case of T ε,∆τ , the corollary provided below holds.
According to Corollaries 18 and 27, the dynamical systems defined as the triplets (M, T ε,∆τ , µ) and (M ′ , T ε,∆τ , µ) have the mixing property where µ corresponds to the Lebesgue measure. In general, the mixing property yields a relaxation process. We adapted this general scenario to our case.
According to Gallavotti [2,8], T ε,∆τ or T ε,∆τ , which has Anosov properties, has a unique probability distribution (SRB distribution) on phase space M or M ′ , respectively. The Lebesgue measure µ is an invariant measure for T ε,∆τ on M and T ε,∆τ on M ′ . In addition, it is obvious that for every measurable partition ζ subordinate to an unstable manifold, the conditional measures of the Lebesgue measure {µ ζ x } are absolutely continuous with respect to the Lebesgue measure. Then, the uniform distribution derived from the Lebesgue measure is the SRB distribution [30].
Convergence to the uniform distribution is proven for (M, T ε,∆τ , µ) and (M ′ , T ε,∆τ , µ), as described above for any initial density function defined except for a zero volume set [8] when the condition (21) is satisfied. In particular, the map T ε,∆τ has time-reversal symmetry. Thus, for the map T ε,∆τ , which satisfies the condition of timereversal symmetry and measure preserving, convergence to the uniform distribution was proven.
In addition to this, correlation functions decay exponentially [2,8,9,31,32]. In other words, in the dynamical system (M, T ε,∆τ , µ) in the regime, relaxation occurs, such that where ρ represents a density function on M and η represents the one-dimensional uniform distribution defined on I δ N,∆τ . In the dynamical system (M ′ , T ε,∆τ , µ), for any orbit {(p 1 (n), q 1 (n))} n∈Z of T ε,∆τ , there exists at least one corresponding orbit {(s n , t n )} n∈Z of T ε,∆τ as the map T ε,∆τ is topologically semi-conjugate with T ε,∆τ . In addition, from Theorem B, any initial density function relaxes to the uniform distribution.

Lyapunov exponent
Let us discuss the Lyapunov exponent of T ε,∆τ . In the following, we consider the case in which N is large enough so that one can identify I δ N,∆τ as − 1 2 , 1 2 and the case in which the condition (21) is satisfied. As the Lebesgue measure is the SRB measure, one can obtain from the Pesin identity the Kolmogorov-Sinai (KS) entropy h KS ( T ε,∆τ ) (entropy per unit time) [33] for almost all initial points as where h KS ( T ε,∆τ ) is positive because T ε,∆τ is an Anosov diffeomorphism. According to Kac [6] and Chrikov [34], it cannot be said that the relaxation always occurs monotonically even though the system has the mixing property. However, there are cases in which by assuming appropriate conditions, the KS entropy is consistent with the time derivative of the entropy [10,11,12].
In calculating h KS ( T ε,∆τ ), one needs the eigenvalue corresponding to the unstable direction, expressed by γ(t), t ∈ I δ N,∆τ with where h KS ( T ε,∆τ ) is equivalent to the positive Lyapunov exponent λ(ε, ∆τ ) owing to the Pesin identity. To obtain an explicit formula of the Lyapunov exponent, we substituted (118) into det D T ε,∆τ | E u .
As it is not necessary that such expressions (119) and (120) are simple enough, we made the following assumption so that an explicit expression could be analytically obtained. (Assumption): One can obtain the equivalent value of (116) if one substitutes cos 2 (πt) = 1/2 into cos 2 (πt) in Equations (119)  , ε < 0.
(124) Figure 3 shows the comparison between the numerical result and analytical formula of the Lyapunov exponent. It is clear that the numerical result is consistent with the analytical formula.

Conclusion
On a microscopic scale, it has been proven that the map T ε,∆τ which has time-reversal symmetry without scatterers, has Anosov properties for ε < 0 or 2 π(∆τ ) 2 < ε namely, the dynamical system derived from the Hamiltonian without scatterers has the mixing property (ergodic).
Further, in the sense of distribution functions, we also have proven that the KS entropy is positive for almost all initial points and that the Lebesgue measure is a unique SRB measure in the parameter ranges. Thus, this indicates that a density function converges to a uniform distribution in the sense of mixing and that the time derivative of the Boltzmann entropy is positive after the system has been mixed sufficiently. This result has assured that dynamical systems can be handled probabilistically or statistically using the uniform distribution.
In summary, the convergence of initial density functions to the uniform distribution and positivity of the entropy rate were proven rigorously in our model derived from the Hamiltonian system with time-reversal symmetry for the parameter ranges ε < 0 or 2 π(∆τ ) 2 < ε. From these results, it can be claimed that the convergence of initial density functions to the uniform distribution and positivity of the entropy rate are compatible with microscopic time reversibility. Thus, from our analysis, a "macroscopic unidirection" can be generated using the internally chaotic mechanism in Hamiltonian dynamical systems with time-reversal symmetry even though the systems do not have external artifacts such as walls, collisions, and scatterers.