# Electrical current in Sinai billiards under general small forces

Nikolai Chernov  Department of Mathematics, University of Alabama at
Birmingham, Birmingham, AL 35294<EMAIL_ADDRESS>,  Hong-Kun Zhang  and
Pengfei Zhang  Department of Mathematics and Statistics, UMass Amherst,
Amherst, MA 01002<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The Lorentz gas of $\mathbb{Z}^{2}$-periodic scatterers (or the so called
Sinai billiards) can be used to model motion of electrons on an ionized medal.
We investigate the linear response for the system under various external
forces (during both the flight and the collision). We give some
characterizations under which the forced system is time-reversible, and derive
an estimate of the electrical current generated by the forced system.
Moreover, applying Pesin entropy formula and Young dimension formula, we get
several characterizations of the non-equilibrium steady state of the forced
system.

##  1\. Introduction

Lorentz gas is a popular model in mathematical physics introduced in 1905,
(see [Lor]), in studying the motions of a point-particle or a gas of particles
(electrons) in a metallic conductor. Here we consider a two-dimensional
periodic Lorentz gas, that is, a particle moves on the plane and bounces off a
$\mathbb{Z}^{2}$-periodic ray of scatterers (ions). In this case the dynamics
reduces to a dynamical billiard system on the 2-D torus $\mathbb{T}^{2}$,
generated by a billiard moving freely until it bounces off the scatterers.
More precisely, let $\mathbf{B}_{1},\cdots,\mathbf{B}_{s}$ be open convex
domains on $\mathbb{T}^{2}$ with mutually disjoint closures, and
$Q=\mathbb{T}^{2}\backslash\bigcup\mathbf{B}_{i}$. Moreover, we assume that
the boundary of each $\mathbf{B}_{i}$ is $C^{3}$ smooth with non-vanishing
curvature.

The study of classical billiard dynamics was originated in 1970 by Sinai. This
model is a Hamiltonian system, so it preserves the kinetic energy
$E=\frac{1}{2}\|\mathbf{p}\|^{2}$. Therefore we can restrict the billiard flow
to a 3-D submanifold
$\mathcal{M}_{0}=\\{(\mathbf{q},\mathbf{p}):E(\mathbf{q},\mathbf{p})=c\\}$, on
which the Liouville measure is also preserved by the flow. Moreover there is a
2-D global cross-section $M_{0}\subset\mathcal{M}_{0}$, which consists of the
post-collision vectors:
$M_{0}=\\{(\mathbf{q},\mathbf{p})\in\mathcal{M}_{0}:\mathbf{q}\in\partial
Q,\mathbf{p}\cdot\mathbf{n}(\mathbf{q})\geq 0\\}$. The Poincaré map
$T_{0}:M_{0}\to M_{0}$ preserves a smooth measure
$d\nu_{0}=\text{cst}\cdot\cos\varphi drd\varphi$, where $\varphi$ is the angle
from the normal vector $\mathbf{n}(\mathbf{q})$ and $\mathbf{p}$. The map
$T_{0}$ and the flow $\Phi_{0}$ have been shown to be uniformly hyperbolic and
Bernoulli [Si70, GO74], and many statistical properties have been well
understood and are proved, see [BSC91, You98, Ch99], and the references
therein.

Recently, much attention is shifting to the investigation of statistical
properties of nonequilibrium billiards. Nonequilibrium phenomena are
characterized by the action of external forces or boundary conditions for
transport equations that change the system and generate a steady process that
can be measured–mass transfer, energy (heat) transfer, charge transfer
(electrical current), entropy production, or others. The laws in equilibrium
statistical physics are better understood and proved or almost proved in quite
a few cases. However, the apparatus of equilibrium statistical mechanics still
relies largely on heuristic statements based on numerical results, and only
very few models have been studied with sufficient mathematical rigor. The main
difficulty is that nonequilibrium billiards have singularities and unbounded
derivatives, they usually do not preserve smooth measures, and their evolution
is described by steady states characterized by singular invariant measures, of
which relatively little is known in general. In addition, since Gibbs entropy
is invariant under a Hamiltonian time evolution, the study of entropy increase
(the second law of thermodynamics) in nonequilibrium systems is far from
straightforward [Ru99]. One of the first nonequilibrium physical models that
were studied rigorously is the periodic Lorentz gas with a constant electrical
field by Chernov, Eyink, Lebowitz and Sinai [CELS93a, CELS93b] and the famous
Ohm’s law was proved for that case. Similar studies on special nonequilibrium
dynamics were conducted by Bunimovich and Spohn [BS94], Gallavotti [G95],
Ruelle [Ru99], and others.

Here we investigate some physical laws for Sinai billiards (or periodic
Lorentz gases) under general external forces. Let $\mathbf{q}=(x,y)$ be the
position of a particle in a billiard table
$Q:=\mathbb{T}^{2}\setminus\cup_{i}\mathbf{B}_{i}$, and
$\mathbf{p}=(\dot{x},\dot{y})$ be the velocity vector. We add two types of
forces to the system in the following steps:

  1. (1)

(during the flight) Let $\mathbf{F}=\mathbf{F}(\mathbf{q},\mathbf{p})$ be a
stationary external force on $Q$. The forced billiard flow is governed by the
following differential equation between collisions:

$\left\\{\begin{array}[]{ll}\dot{\mathbf{q}}=\mathbf{p},\\\
\dot{\mathbf{p}}=\mathbf{F},\end{array}\right.$ (1.1)

where the dot derivative refers to differentiate with respect to the time $t$.

  2. (2)

(at the moment of collision) Let $\mathbf{G}$ be an external twisting force
spreading on $\partial Q$ that acts on each incoming trajectory right after
its elastic collision with $\partial Q$:

$(\mathbf{q}^{+}(t_{i}),\mathbf{p}^{+}(t_{i}))=\mathbf{G}(\mathbf{q}^{-}(t_{i}),\mathcal{R}\mathbf{p}^{-}(t_{i})),$
(1.2)

where
$\mathcal{R}\mathbf{p}^{-}(t_{i})=\mathbf{p}^{-}(t_{i})+2(\mathbf{n}(\mathbf{q}^{-})\cdot\mathbf{p}^{-})\mathbf{n}(\mathbf{q}^{-}))$
is the usual elastic reflection operator, $\mathbf{n}(\mathbf{q})$ is the unit
normal vector to the billiard wall $\partial Q$ at $\mathbf{q}$ pointing
inside of the table $Q$, and $\mathbf{q}^{-}(t_{i}),\mathbf{p}^{-}(t_{i})$,
$\mathbf{q}^{+}(t_{i})$ and $\mathbf{p}^{+}(t_{i})$ refer to the incoming and
outgoing position and velocity vectors, respectively.

Note that the twisting force $\mathbf{G}$ changes not only the outgoing
velocity of the billiard, but also the position of the billiard along the
boundary $\partial Q$. The change in velocity can be thought of as a kick,
while a change in position can model a slip along the boundary at collision.
These forces indicate that the system will experience nonelastic reflections.

The type of forces we consider here are quite common in many physical models.
For example we can have a potential function $U$ on $Q$, or an electromagnetic
field (when the moving billiard is an electron), see [CELS93a, CELS93b, Ch01,
Ch08]. The twisting force right after the collision is also closely related to
real-world models, especially the so called soft scatterers, see [Ba88, Kn89].
Recall that the unforced system can be viewed as a Hamiltonian system under a
potential function $U$ given by $U(\mathbf{q})=0$ if $\mathbf{q}\in Q$, and
$+\infty$ on the scatterers. In this case, the scatterers $B_{i}$ are said to
be rigid (or hard). Now if we replace $U$ by some finite potential function on
each $B_{i}$, then the resulting $B_{i}$ is a soft one: a running billiard
will climb up $B_{i}$ and then exit very soon. Different shapes of the
scatterer result in different enter-exit relations. If we view what happen on
the scatterers as a black box, the effect can be understood as a twisting
force $\mathbf{G}$ right after the elastic collision. Clearly this kind of
twisting forces not only preserve the tangent collisions, and are also time-
reversible. In this way we get a uniform treatment of various softened
scatterers.

In [Ch01, Ch08], Chernov considered billiards under small external forces
$\mathbf{F}$ between collisions, and proved several ergodic and statistical
properties of the SRB measure for the perturbed billiard system. In [DC09]
Dolgoyat and Chernov put a constant electric field on Lorentz gases with
infinite horizon and got various characterizations of the steady state
electric current generated by the forced system. The systems with some simple
twist forces were considered in [Zh11], assuming that $\mathbf{G}$ depends on
and affects only the velocity, not the position. The Green-Kubo type formula
was proved, and it was shown that the current generated by the forced flow is
closely related to the strength of the force. Very recently, Chernov and
Korepanov investigated in [CK13] the linear response for Sinai billiards under
external forces (without twisting). In this paper, we consider the dynamics of
the Sinai billiard on the table $Q$, but subject to more general forces
$\mathbf{P}=(\mathbf{F},\mathbf{G})$ both during flight and at collisions. We
characterized certain properties of the SRB measure for the forced systems by
obtaining the Pesin’s K-S entropy formula and Young’s expression for the
fractal dimension. Moreover, we also prove rigorously the Green-Kubo like
formula and investigate the linear response formula.

Structure of the paper. This paper is organized as follows. In §2 we list the
main assumptions on the external forces $\mathbf{P}=(\mathbf{F},\mathbf{G})$,
introduce some basic notations and propositions of our systems. In §3 we state
the main theorems about the linear responses and statistical properties of our
forced billiard system. We also list quite a few examples of the external
forces and study the current generated by these forced systems. Then we divide
the analysis of the forced systems into two steps: in §3 we study the effect
of the force $\mathbf{F}$ during the flight, and in §4 we add the twisting
effect and conclude the proof of our main theorems.

##  2\. Main results

###  2.1. Assumpsions

In this subsection we first state the assumptions on the model, which combine
the assumptions in [Ch01, Zh11]. Let $\mathbf{P}=(\mathbf{F},\mathbf{G})$,
where $\mathbf{F}$ and $\mathbf{G}$ are the two external forces during the
flight and right after the reflection, respectively. Let $\Phi_{\mathbf{P}}$
be the induced billiard flow on $Q$.

(A1) (Invariant space) _The forced flow $\Phi_{\mathbf{P}}$ preserve a smooth
function $\mathcal{E}(\mathbf{q},\mathbf{p})$, such that the level surface
$\mathcal{M}:=\\{\mathcal{E}(\mathbf{q},\mathbf{p})=c\\}$ is a compact 3-D
manifold, for some $c>0$. Moreover, $\|\mathbf{p}\|>0$ on $\mathcal{M}$, and
for each $\mathbf{q}\in Q$ and $\mathbf{p}\in S^{1}$, the ray
$\\{(\mathbf{q},t\mathbf{p}),t>0\\}$ intersects the manifold $\mathcal{M}$ in
exactly one point._

Assumption (A1) specifies an additional integral of motion, so that we only
consider the restricted systems on a compact phase space. For example, we can
add a Gaussian thermostat (a heat bath) to the system such that the billiard
moves at a constant speed (constant temperature if there are a large number of
particles). Then $\mathcal{M}:=\\{\|\mathbf{p}\|=c\\}$ is an invariant compact
level set.

Under the assumption (A1), the speed $p=\|\mathbf{p}\|$ of the billiard along
any typical trajectory on $\mathcal{M}$ at time $t$ satisfies

$0<p_{\min}\leq p(t)\leq p_{\max}<\infty,$

for some constants $p_{\min}\leq p_{\max}$. Moreover, $\mathcal{M}$ admits a
global coordinate system $\\{(x,y,\theta):(x,y)\in Q,0\leq\theta<2\pi\\}$,
where $\theta$ is the angle between $\mathbf{p}$ and the positive $x$-axis. In
particular, the speed $p=\|\mathbf{p}\|$ on $\mathcal{M}$ can be represented
as a function $p=p(x,y,\theta)$. Then the velocity $\mathbf{p}$ at
$\mathbf{q}$ is given by $\mathbf{p}=p\mathbf{v}$, where
$\mathbf{v}=(\cos\theta,\sin\theta)$ is the unit vector in the direction of
$\mathbf{p}$. So Eq. (1.1) of the dynamics between collisions can be rewritten
as

$\dot{p}\mathbf{v}+p\dot{\mathbf{v}}=\mathbf{F}.$ (2.1)

Multiplying $\mathbf{v}$ to both sides of (2.1) using dot product and cross
product respectively, we then get

$\dot{p}=\mathbf{v}\cdot\mathbf{F},\,\,\,\,\,\text{ and
}\,\,\,\,\,\,p\mathbf{v}\times\dot{\mathbf{v}}=\mathbf{v}\times\mathbf{F}.$
(2.2)

Therefore, the equations in (1.1) have the following coordinate
representations that, at any $(x,y,\theta)\in\mathcal{M}$,

$\left\\{\begin{array}[]{ll}\dot{x}=p\cos\theta,\\\ \dot{y}=p\sin\theta,\\\
\dot{\theta}=(-F_{1}\sin\theta+F_{2}\cos\theta)/p.\end{array}\right.$ (2.3)

Consider the trajectory $\tilde{\gamma}\subset\mathcal{M}$ of the flow passing
through the point $(x,y,\theta)\in\mathcal{M}$, which projects down to a
smooth curve $\gamma\subset Q$. Let $h=h(x,y,\theta)$ be the (signed)
geometric curvature of $\gamma$ at the base point $(x,y)\in Q$. Then we have
that

$h(x,y,\theta)=\pm\frac{\|\dot{\mathbf{q}}\times\ddot{\mathbf{q}}\|}{\|\dot{\mathbf{q}}\|^{3}}=\pm\frac{\|\mathbf{v}\times\mathbf{F}\|}{p^{2}}=\frac{-F_{1}\sin\theta+F_{2}\cos\theta}{p^{2}},$
(2.4)

where the sign should be chosen accordingly. Then combining with (2.3), we
have

$\dot{\theta}=ph.$ (2.5)

Note that the angle $\theta=\theta(t)$ experiences a discontinuity at the
times of reflection. That is, it changes from $\theta^{-}$ to $\theta^{+}$. In
the elastic collision case, all other quantities ($x$, $y$ and $p$) stay the
same. For example the speed $p(x,y,\theta^{+})=p(x,y,\theta^{-})$ (here
$(x,y)\in\partial Q$). Under the twisting forces, all quantities are subject
to change.

For any phase point $(\mathbf{q},\theta)\in\mathcal{M}$ for the flow, let
$\tau(\mathbf{q},\theta)$ be the time for the trajectory from
$(\mathbf{q},\theta)$ to its next non-tangential collision at $\partial Q$.

(A2) (Finite horizon) _There exist $\tau_{\max}>\tau_{\min}>0$ such that free
paths between successive reflections are uniformly bounded:
$\tau_{\min}\leq\tau(\mathbf{q},\theta)\leq\tau_{\max}$, for all
$(\mathbf{q},\theta)\in\mathcal{M}$ with $\mathbf{q}\in\partial Q$. In
addition, the curvature $\mathcal{K}(r)$ of the boundary $\partial Q$ is also
uniformly bounded for all $r\in\partial Q$._

Assumption (A2) implies that there exists a 2-D global cross-section, the
post-collision space $M$, of the perturbed billiard flow
$(\mathcal{M},\Phi_{\mathbf{P}})$:
$M=\\{(\mathbf{q},\mathbf{p})\in\mathcal{M}:\mathbf{q}\in\partial
Q,\mathbf{p}\cdot\mathbf{n}(\mathbf{q})\geq 0\\}$, which consists all outgoing
vectors in $\mathcal{M}$ based at the boundary of the billiard table $Q$.
Denote by $T_{\mathbf{P}}:M\to M$ the Poincaré map induced by the forced flow
$\Phi_{\mathbf{P}}$ on $\mathcal{M}$. Moreover, the 2-D space $M$ can be
parameterized by $\mathbf{x}=(r,s)$, where $r$ is the arclength parameter of
$\partial Q$ oriented clockwise, and a new parameter $s=\sin\varphi$, where
$\varphi$ is the angle formed by the outgoing vector $\mathbf{p}$ and the
normal vector $\mathbf{n}(\mathbf{q})$. This coordinate system has the
advantage that the Lebesgue measure $d\mu_{0}={\rm Cst}\,dr\,ds$ coincides
with the measure $d\nu_{0}={\rm Cst}\cos\varphi dr\,d\varphi$. Using this new
coordinate system, the collision time can be written as
$\tau(\mathbf{x})=\tau(r,s)$, and the twist force $\mathbf{G}$ at the
collision can be reformulated as
$(\bar{r},\bar{s})=\mathbf{G}(r,s)=(r,s)+(g^{1}(r,s),g^{2}(r,s))$.

(A3) (Smallness of the external forces). _There exists $\varepsilon>0$ small
enough, such that the forces $\mathbf{P}=(\mathbf{F},\mathbf{G})$ satisfy_

$\|{\mathbf{F}}\|_{C^{1}}<\varepsilon,\|\mathbf{G}-\mathrm{Id}_{M}\|_{C^{1}}<\varepsilon.$

_Moreover, we assume that $\mathbf{G}$ preserves tangential collisions:
$\mathbf{G}(r,\pm 1)=(r,\pm 1)$. In other words, $g^{i}(r,\pm 1)=0$ for each
$i=1,2$. _

In particular, the singularity set of $T^{-1}_{\mathbf{P}}$ is the same as
that of untwisted map $T^{-1}_{\mathbf{F}}$.

Let $\mathcal{I}:\mathcal{M}\to\mathcal{M}$ be the reversal transformation
(also called involution), which is defined by
$\mathcal{I}(x,y,\theta)=(x,y,\pi+\theta)$. Let
$\Phi_{t}:\mathcal{M}\to\mathcal{M}$ be a general flow. The reversed flow of
$\Phi$ is defined by $\Phi_{t}^{-}=\mathcal{I}\circ\Phi_{-t}\circ\mathcal{I}$.
Then the flow $\Phi$ is said to be time-reversible, if
$\Phi_{t}^{-}=\Phi_{t}$. It is well known that the unforced billiard flow is
time-reversible.

(A4) (Time-reversibility) _Both forces $\mathbf{F}$ and $\mathbf{G}$ are
stationary, and the forced billiard flow $\Phi_{\mathbf{P}}$ is time-
reversible._

Let $\varepsilon>0$, $\tau_{\ast}\in(0,1)$ and
$\mathcal{F}(Q,\tau_{\ast},\varepsilon)$ be the collection of all forced
billiard maps defined by the dynamics (1.1) and (1.2) under the external
forces $\mathbf{P}=(\mathbf{F},\mathbf{G})$ that satisfying the assumptions
(A1)–(A4) with
$\tau_{\ast}\leq\tau_{\min}\leq\tau_{\max}\leq\tau_{\ast}^{-1}$. The following
lemma was proved by Demers and Zhang in [DZ11]:

######  Lemma 2.1.

Each map $T\in\mathcal{F}(Q,\tau_{\ast},\varepsilon)$ preserves a unique SRB
measure (the nonequilibrium steady state) $\mu_{T}$ that is mixing, Bernoulli
and positive on open sets in $M$. Let $\mathcal{H}$ be the collection of all
piecewise Hölder continuous functions on $M$ where the discontinuities occur
on the singularity sets $T$. Then

  1. a.

(Equidistribution) for any $f\in\mathcal{H}$, $T^{n}\mu_{0}(f)\to\mu_{T}(f)$
at an exponential rate;

  2. b.

(Decay of Correlations) for any $f,g\in\mathcal{H}$, $\mu_{0}(f\circ
T^{n}\cdot g)\to\mu_{T}(f)\cdot\mu_{0}(g)$ at an exponential rate.

  3. c.

(Central Limit Theorem) for any $f\in\mathcal{H}$, $f_{n}=f+f\circ
T+\cdots+f\circ T^{n-1}$, then
$\frac{f_{n}}{\sqrt{n}}\Rightarrow\mathcal{N}(\mu_{T}(f),\sigma^{2}_{f})$,
where $\sigma^{2}_{f}=\sum_{n\in\mathbb{Z}}\mathcal{C}_{f,f}(T^{n})$. Here the
convergence means that the distributions of $\frac{f_{n}}{\sqrt{n}}$ converge
to the normal distribution $\mathcal{N}(\mu_{T}(f),\sigma^{2}_{f})$.

Moreover, the flow $\Phi$ preserves a mixing SRB measure $\hat{\mu}_{T}$ with
similar properties.

###  2.2. Main results

In this section we state the main results of this paper, the properties of
forced billiard systems under the assumptions (A1)–(A4). More precisely, let
$\varepsilon>0$ be small enough, $\tau_{\ast}\in(0,1)$ and
$\mathbf{P}=(\mathbf{F},\mathbf{G})$ be an $\varepsilon$-small external force
pair such that $T_{\mathbf{P}}\in\mathcal{F}(Q,\tau_{*},\varepsilon)$,
$\mu_{\mathbf{P}}$ be the SRB measure on $M$ given in Theorem 2.1. Let
$\widetilde{Q}$ be the $\mathbb{Z}^{2}$-periodic table on the plane,
$\widetilde{T}_{\mathbf{P}}$ be the forced collision map on the
$\mathbb{Z}^{2}$-periodic collision space $\widetilde{M}$, and

$\widetilde{\boldsymbol{\Delta}}_{\mathbf{P}}(\mathbf{x})=(\widetilde{\Delta}_{x,\mathbf{P}},\widetilde{\Delta}_{y,\mathbf{P}})=\tilde{\pi}\circ\widetilde{T}_{\mathbf{P}}(\mathbf{x})-\tilde{\pi}(\mathbf{x})$
(2.6)

be the displacement vector between collisions, where $\tilde{\pi}$ is the
projection from $\widetilde{M}$ to the base point in $\widetilde{Q}$. It is
easy to see that that $\widetilde{\boldsymbol{\Delta}}_{\mathbf{P}}$ is
$\mathbb{Z}^{2}$-periodic, which induces a displacement function on the
collision space $M$, say ${\boldsymbol{\Delta}}_{\mathbf{P}}$. Our first
theorem describes estimations on the current for the discrete system.

######  Theorem 2.2.

(a) The discrete-time steady state electrical current is well-defined and
given by:

$\mathbf{J}_{\mathbf{P}}=\lim_{n\to\infty}\tilde{\mathbf{q}}_{n}/n=\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}}).$
(2.7)

(b) The current ${\mathbf{J}_{\mathbf{P}}}$ satisfies
$\mathbf{J}_{\mathbf{P}}=\varepsilon{\boldsymbol{\sigma}}+o(\varepsilon)$,
where ${\boldsymbol{\sigma}}=(\sigma_{x},\sigma_{y})$ is uniformly bounded and
satisfies

$\sigma_{a}=\frac{1}{2}\mu_{0}(\Delta_{a,\mathbf{P}}\cdot
H)+\sum_{k=1}^{\infty}\mu_{0}[(\Delta_{a,\mathbf{P}}\circ T^{k}_{0})\cdot
H],\quad a\in\\{x,y\\},$ (2.8)

where
$H(r,s)=(2-\exp(\int_{0}^{\tau_{\mathbf{F}}(r,s)}ph_{\theta}\,dt)-\mathcal{J}_{\mathbf{G}}(r_{1},s_{1}))/\varepsilon$
is a uniformly bounded function, and $\mathcal{J}_{\mathbf{G}}$ is the
Jacobian of the twisting force $\mathbf{G}$.
(c) As $n\to\infty$,

$\frac{\tilde{\mathbf{q}}_{n}-n\mathbf{J}_{\mathbf{P}}}{\sqrt{n}}\Rightarrow\mathcal{N}(0,\mathbf{D}_{\mathbf{P}}),$
(2.9)

where $\mathbf{D}_{\mathbf{P}}$ is the discrete-time diffusion matrix of the
Lorentz particle, which is given by:

$\mathbf{D}_{\mathbf{P}}=\sum_{n=-\infty}^{\infty}\left[\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}}\circ
T^{n}_{\mathbf{P}}\otimes{\boldsymbol{\Delta}}_{\mathbf{P}})-\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}})\otimes\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}})\right]$
(2.10)

(d) $\mathbf{D}_{\mathbf{P}}$ is continuous with respect to the size of the
force pair $\mathbf{P}$ at $\mathbf{P}=0$:

$\mathbf{D}_{\mathbf{P}}=\mathbf{D}_{0}+o(1).$ (2.11)

In particular, the generated current $\mathbf{J}_{\mathbf{P}}$ is comparable
to the size of the external force $\mathbf{P}$, where ${\boldsymbol{\sigma}}$
resembles the electric conductivity of the forced system. Moreover, the drift
effect is sub-linear. One may wonder if we could use the linear approximates
$\int_{0}^{\tau_{\mathbf{F}}(r,s)}ph_{\theta}\,dt$ for the function $H$ in
(b). However, this may destroy the convergence of the series (2.8) (see Remark
4 for more discussions). However, we do use the linear term for all physical
models with a Gaussian thermostat (see Proposition 2.6, 2.7 and Corollary
2.8).

The corresponding results for the continuous-time forced system is provided by
the following theorem.

######  Theorem 2.3.

Suppose a particle move in the domain $\widetilde{Q}$ under the external force
$\mathbf{P}=(\mathbf{F},\mathbf{G})$.
(a) The steady state current generated by $\Phi_{\mathbf{P}}$ is well-defined
and given by

$\hat{\mathbf{J}}_{\mathbf{P}}=\lim_{t\to\infty}\tilde{\mathbf{q}}(t)/t=\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}})/\mu_{\mathbf{P}}(\tau_{\mathbf{P}}).$
(2.12)

(b) The current $\hat{\mathbf{J}}_{\mathbf{P}}$ satisfies

$\hat{\mathbf{J}}_{\mathbf{P}}=\varepsilon\cdot\frac{{\boldsymbol{\sigma}}}{\bar{\tau}_{\mathbf{P}}}+o(\varepsilon).$
(2.13)

(c) As $t\to\infty$,

$\frac{\tilde{\mathbf{q}}(t)-\hat{\mathbf{J}}_{\mathbf{P}}t}{\sqrt{t}}\Rightarrow\mathcal{N}(0,\hat{\mathbf{D}}_{\mathbf{P}}),$
(2.14)

where $\hat{\mathbf{D}}_{\mathbf{P}}$ is the continuous-time diffusion matrix
of the Lorentz particle.
(d) The diffusion matrix is continuous with respect to the size of force pair
$\mathbf{P}$ at $\mathbf{P}=0$:

$\hat{\mathbf{D}}_{\mathbf{P}}=\hat{\mathbf{D}}_{0}+o(1)$ (2.15)

We also give some characterizations of the nonequilibrium steady state
$\mu_{\mathbf{P}}$ of the forced system $T_{\mathbf{P}}$:

######  Theorem 2.4.

(1) The measure $\mu_{\mathbf{P}}$ satisfies the Pesin entropy formula:

$h_{\mu_{\mathbf{P}}}(T_{\mathbf{P}})=\lambda_{\mathbf{P}}^{u},$ (2.16)

where $\lambda^{s}_{\mathbf{P}}<0<\lambda^{u}_{\mathbf{P}}$ are the Lyapunov
exponents for the measure $\mu_{\mathbf{P}}$ and
$h_{\mu_{\mathbf{P}}}(T_{\mathbf{P}})$ is the metric entropy for
$(T_{\mathbf{P}},\mu_{\mathbf{P}})$;
(2) $\mu_{\mathbf{P}}$ satisfies Young’s dimension formula:

$\mathrm{HD}(\mu_{\mathbf{P}})=h_{\mu_{\mathbf{P}}}(T_{\mathbf{P}})\left(\frac{1}{\lambda^{u}_{\mathbf{P}}}-\frac{1}{\lambda^{s}_{\mathbf{P}}}\right),$
(2.17)

where $\mathrm{HD}(\mu_{\mathbf{P}})$ is the Hausdorff dimension of the
measure $\mu_{\mathbf{P}}$.
(3) Let $h_{0}=h_{\mu_{0}}(T_{0})$ be the metric entropy of $T_{0}$, then

$\displaystyle\mathrm{HD}(\mu_{\mathbf{P}})$
$\displaystyle=2-\varepsilon^{2}\cdot\frac{\sigma^{2}_{H}}{2h_{0}}+o(\varepsilon^{2}),\>\text{
where }\>\sigma^{2}_{H}=\sum_{k=-\infty}^{\infty}\mu_{0}[H\circ T_{0}^{k}\cdot
H].$ (2.18)

It follows from Eq. (2.18) that $1<\mathrm{HD}(\mu_{\mathbf{P}})<2$ for some
small external forces $\mathbf{P}$. So $\mu_{\mathbf{P}}$ is singular with
respect to the Lebesgue measure and admits a fractal structure.

Let $\hat{\mu}_{\mathbf{P}}$ be the corresponding SRB measure preserved by the
forced flow $\Phi^{t}_{\mathbf{P}}$ on $\mathcal{M}$. The metric entropy of
the measure $\hat{\mu}_{\mathbf{P}}$ is given by
$h_{\hat{\mu}_{\mathbf{P}}}(\Phi_{\mathbf{P}}):=h_{\mu_{\mathbf{P}}}(T_{\mathbf{P}})/\mu_{\mathbf{P}}(\tau_{\mathbf{P}})$
and the fractal dimension
$\mathrm{HD}(\hat{\mu}_{\mathbf{P}})=\mathrm{HD}(\mu_{\mathbf{P}})+1$.
Therefore similar formulas in Theorem 2.4 hold for $\hat{\mu}_{\mathbf{P}}$.

###  2.3. Applications

Next we provide some example of reversible external forces and give the
generated currents by the forced billiard systems.

Example 1. Conservative forces on $Q$.

We consider a conservative force $\mathbf{F}=-\nabla U(\mathbf{q})=-\langle
U_{x},U_{y}\rangle$, where $U(\mathbf{q})$ is a (small) potential function on
$Q$. The induced billiard system $\Phi_{\mathbf{F}}$ preserves the total
energy $E=\frac{1}{2}p^{2}+U(\mathbf{q})$ of the system and hence satisfies
Assumption (A1). We restrict the dynamics to a energy level
$\mathcal{M}=\\{(\mathbf{q},\mathbf{p}):E(\mathbf{q},\mathbf{p})=1/2\\}$. In
particular the speed function $p^{2}(x,y,\theta)=1-2U(\mathbf{q})$ is
independent of $\theta$, say $p=p(x,y)$. It is well known that the billiard
flow $\Phi_{\mathbf{F}}$ is time-reversible (see also Lemma 3.1 and Remark 2).
Moreover, it is easy to see that the generating vector field $X=\langle
p\cos\theta,p\sin\theta,ph\rangle$ is divergence-free:

$\text{div
}X=p_{x}\cos\theta+p_{y}\sin\theta+(U_{x}\cos\theta+U_{y}\sin\theta)/p=\frac{1}{p}(\dot{p}-\mathbf{v}\cdot\mathbf{F})=0.$
(2.19)

In particular, the flow $\Phi_{\mathbf{F}}$ preserves the Lebesgue measure $m$
on the energy level $\mathcal{M}$ and $\hat{\mu}_{\mathbf{F}}=m$. Since there
is no slip after the collision, the current is indeed zero (see the related
discussion in Remark 5):

$\mathbf{J}_{\mathbf{F}}=m(\mathbf{p})=\int_{Q}p(x,y)(\int_{S^{1}}\mathbf{v}d\theta)dxdy=0.$
(2.20)

Example 2. Conservative twists and soft scatterers.

Let $\mathbf{B}$ be a small scatterer on $Q$, $U$ a potential function on
$\mathbb{T}^{2}$ such that $U=0$ on $Q\backslash\mathbf{B}$ and $U>0$ on
$\mathbf{B}$. We consider the energy surface
$E(\mathbf{q},\mathbf{p})=\frac{1}{2}\|\mathbf{p}\|^{2}+U(\mathbf{q})=1$. So a
running billiard may climb up the scatterer, and regain its full kinetic
energy whenever it exits that scatterer. If we view what happened on the
scatterer as a black box, the reduced dynamics is close to the unforced system
during its free flight. The only difference is that the exit location and
direction are different from the elastic collision (corresponding to
$U=+\infty$ on $\mathbf{B}$). Therefore, the effect is equivalent to applying
a twist force $\mathbf{G}$ right after the collision. Clearly the perturbed
flow $\Phi$ is time-reversible. Moreover, by embedding $\Phi$ into the real
Hamiltonian flow on the ambient torus, we see that $\Phi_{\mathbf{G}}$
preserves the canonical space $\mathcal{M}_{0}$ and the Liouville measure on
$\mathcal{M}_{0}$. In particular its current $\mathbf{J}_{\mathbf{G}}=0$.

To get an intuition of the vanishing currents, we consider a special case
explicitly.

######  Proposition 2.5.

Suppose all the scatterers $\mathbf{B}_{i}$ are all round disks centered at
$\mathbf{q}_{i}=(x_{i},y_{i})$ of radius $\delta_{i}$, the potential is given
by
$U(\mathbf{q})=\varepsilon^{-1}\cdot(\delta_{i}-\|\mathbf{q}-\mathbf{q}_{i}\|)$
on $\mathbf{B}_{i}$, and $U=0$ outside the scatterers. Then the forced flow
has vanishing current.

###### Proof.

In the case it is easy to see that the outgoing direction is the same as the
hard scatterer case, the slip does not depend on where the collision happens:
$\mathbf{G}(r,s)=(r+g_{i}(s),s)$. So the Jacobian
$\mathcal{J}_{\mathbf{G}}\equiv 1$. Therefore the current must be zero. ∎

Example 3. Isokinetic forces.

Next we consider forces $\mathbf{F}$ that are always perpendicular to the
momentum. In this case, the forced billiard flow preserves the kinetic energy.
Without loss of generality, we assume $\|\mathbf{p}\|=1$ (as in the classical
billiards), and reformulate the force as

$\mathbf{F}=F\mathbf{v}^{\perp}=F(x,y,\theta)(-\sin\theta,\cos\theta),$

where $F=F(x,y,\theta)$ is a scale function (may be negative) with
$\|F\|_{C^{1}}<\varepsilon$. By Eq. (2.4), the geometric curvature
$h(x,y,\theta)$ satisfies

$h(x,y,\theta)=F(x,y,\theta)(\sin^{2}\theta+\cos^{2}\theta)/p^{2}=F(x,y,\theta).$
(2.21)

Not all isokinetic forced systems are time-reversible. For example, the system
of an electron moving under a constant magnetic field perpendicular to $Q$ is
not time-reversible.

In the following we assume that $F$ satisfies
$F(x,y,\theta+\pi)=-F(x,y,\theta)$. Then the forced billiard flow
$\Phi_{\mathbf{F}}$ is time-reversible (see Lemma 3.1). A special feature of
these isokinetic forces is that we can use the linear term of $H$ to estimate
the generated current:

######  Proposition 2.6.

The discrete-time steady state current under a general isokinetic force
$\mathbf{F}$ is given by
$\mathbf{J}_{\mathbf{F}}=\varepsilon{\boldsymbol{\sigma}}+o(\varepsilon)$,
where ${\boldsymbol{\sigma}}=(\sigma_{x},\sigma_{y})$ is given by

$\sigma_{a}=-\frac{1}{2}\mu_{0}[\Delta_{a,\mathbf{F}}\cdot\int_{0}^{\tau_{\mathbf{F}}(\mathbf{x})}F_{\theta}\,dt/\varepsilon]-\sum_{k=1}^{\infty}\mu_{0}[(\Delta_{a,\mathbf{F}}\circ
T^{k}_{0})\cdot\int_{0}^{\tau_{\mathbf{F}}(\mathbf{x})}F_{\theta}\,dt/\varepsilon],\quad
a\in\\{x,y\\}.$

Example 4. Electric field with thermostat. We consider an electric field
$\mathbf{E}(\mathbf{q})=(\varepsilon e_{1}(\mathbf{q}),\varepsilon
e_{2}(\mathbf{q}))$ on $\mathbf{q}\in Q$. It may generate a net velocity in
the force direction and keep accelerating the electron. We can modify the
system by adding a constraining force to maintain the system at a constant
temperature (a compact level set, say $\mathcal{E}_{1}=\\{\|\mathbf{p}\|\equiv
1\\}$), and to preserve a steady state on that level. More precisely, the
system of the forced equations on $\mathcal{E}_{1}$ is given by:
$\dot{\mathbf{q}}=\mathbf{p}$, $\dot{\mathbf{p}}=\mathbf{E}-\alpha\mathbf{p}$,
where $\alpha=\mathbf{E}\cdot\mathbf{p}$ is a thermostat. For such systems, we
have the following results.

######  Proposition 2.7.

Let $\mathbf{E}(\mathbf{q})=(\varepsilon e_{1}(\mathbf{q}),\varepsilon
e_{2}(\mathbf{q}))$ be an electric field on $\mathbf{q}\in Q$. The discrete-
time steady state electrical current under $\mathbf{E}$ with thermostat is
given by
$\mathbf{J}_{\mathbf{E}}=\varepsilon{\boldsymbol{\sigma}}+o(\varepsilon)$,
where ${\boldsymbol{\sigma}}=(\sigma_{x},\sigma_{y})$ is given by

$\sigma_{a}=\frac{1}{2}\mu_{0}(\Delta_{a,\mathbf{E}}\cdot\int_{0}^{\tau_{\mathbf{E}}}(e_{1},e_{2})\cdot\mathbf{p}\,dt)+\sum_{k=1}^{\infty}\mu_{0}[(\Delta_{a,\mathbf{E}}\circ
T^{k}_{0})\cdot\int_{0}^{\tau_{\mathbf{E}}}(e_{1},e_{2})\cdot\mathbf{p}\,dt].$

Now we consider a constant electric field $\mathbf{E}=(\varepsilon
e^{0}_{1},\varepsilon e^{0}_{2})$ on the table $Q$. In this case it is easy to
see that
$\int_{0}^{\tau_{\mathbf{E}}(r,s)}(e^{0}_{1},e^{0}_{2})\cdot\mathbf{p}dt=(e^{0}_{1},e^{0}_{2})\cdot{\boldsymbol{\Delta}}_{\mathbf{E}}(r,s)$.
So we have

######  Corollary 2.8.

Let $\mathbf{E}(\mathbf{q})=(\varepsilon e^{0}_{1},\varepsilon e^{0}_{2})$ be
a constant electric field on $\mathbf{q}\in Q$. The discrete-time steady state
electrical current under $\mathbf{E}$ with thermostat is given by
$\mathbf{J}_{\mathbf{E}}=\varepsilon{\boldsymbol{\sigma}}+o(\varepsilon)$,
where ${\boldsymbol{\sigma}}=(\sigma_{x},\sigma_{y})$ is given by

$\displaystyle\sigma_{a}=$
$\displaystyle\frac{1}{2}\mu_{0}(\Delta_{a,\mathbf{E}}\cdot
H)+\sum_{k=1}^{\infty}\mu_{0}[(\Delta_{a,\mathbf{E}}\circ T^{k}_{0})\cdot
H],\quad a\in\\{x,y\\},$

where
$H(r,s)=(e^{0}_{1},e^{0}_{2})\cdot{\boldsymbol{\Delta}}_{\mathbf{E}}(r,s)$.

This is the current formula obtained in [CELS93a].

##  3\. Preliminary properties of $T_{\mathbf{F}}$ under the force
$\mathbf{F}$

We divide our study of the forced system $T_{\mathbf{P}}$ into two steps
according to the nature of the force pair $\mathbf{P}$: the step of a elastic
reflection under the force $\mathbf{F}$, and the twisting by $\mathbf{G}$
right after the collision. In this section we consider the pre-twist step,
that is, the effect of the force $\mathbf{F}$ between one collision. First we
state the time-reversibility of the forced system:

######  Lemma 3.1.

Let $\mathbf{F}$ be an external force on the table $Q$, $p$ and $h$ be the
speed and the curvature functions of the forced billiard flow
$\Phi_{\mathbf{F}}$. Then $\Phi_{\mathbf{F}}$ is time-reversible if and only
if the following conditions hold for any $(x,y,\theta)\in\mathcal{M}$:

$\displaystyle p(x,y,\pi+\theta)$ $\displaystyle=p(x,y,\theta),$ (3.1a)
$\displaystyle h(x,y,\pi+\theta)$ $\displaystyle=-h(x,y,\theta).$ (3.1b)

The proof is straightforward and omitted here. See also [Ch01, P. 209–210] for
detailed discussions.

######  Remark 1.

There is a canonical involution $I$ on the post-collision space $M$, which is
given by $I:M\to M$, $(r,s)\mapsto(r,-s)$. Let $T^{-}_{\mathbf{F}}$ be the
Poincaré map of the reversed flow $\Phi_{\mathbf{F}}^{-}$. It is easy to see
that $T^{-}_{\mathbf{F}}=I\circ T^{-1}_{\mathbf{F}}\circ I$. So time-
reversibility of the forced flow $\Phi_{\mathbf{F}}^{-}=\Phi_{\mathbf{F}}$
implies the time-reversibility of the forced map:
$T_{\mathbf{F}}^{-}=T_{\mathbf{F}}$.

######  Remark 2.

A special case is that the force $\mathbf{F}=\mathbf{F}(\mathbf{q})$ depends
only on the position $\mathbf{q}$. For example, for a particle moving in the
gravitivity field and an electron moving in an electric field, the forces do
not depend on the velocity $\mathbf{p}$. According to Eq. (2.4), we see that
(3.1b) follows from (3.1a). So these forced systems are time-reversible if and
only if $p(x,y,\pi+\theta)=p(x,y,\theta)$ for any
$(x,y,\theta)\in\mathcal{M}$.

######  Lemma 3.2.

Let $m$ be the Lebesgue measure on $M$, $\mathbf{x}=(r,s)$ and
$T_{\mathbf{F}}\mathbf{x}=(r_{1},s_{1})$. Then the Jacobian of
$DT_{\mathbf{F}}$ with respect to the Lebesgue measure $\mu_{0}$ is given by

$\det
D_{\mathbf{x}}T_{\mathbf{F}}=\exp(\int_{0}^{\tau_{\mathbf{F}}(\mathbf{x})}ph_{\theta}dt).$
(3.2)

###### Proof.

Let $X(x,y,\theta)=\langle p\cos\theta,p\sin\theta,ph\rangle$ be the vector
field on $\mathcal{M}$ that generates the flow $\Phi^{t}_{\mathbf{F}}$.
Consider the Lebesgue measure $dm=dx\,dy\,d\theta$ on $\mathcal{M}$. Note that
$m$ is not necessarily invariant under the forced flow $\Phi_{\mathbf{F}}$,
and its rate of change is given by the divergence of the generating vector
field $X$:

$\text{div
}X(x,y,\theta)=p_{x}\cos\theta+p_{y}\sin\theta+p_{\theta}h+ph_{\theta}=(p_{x}\dot{x}+p_{y}\dot{y}+p_{\theta}\dot{\theta})/p+ph_{\theta}=\frac{d\ln
p}{dt}+ph_{\theta}.$

Note that the force billiard flow can also be represented as a suspension of
the forced map $(M,T_{\mathbf{F}},\mu_{0})$ with respect to the roof function
$\tau_{\mathbf{F}}$. In particular the volume $dm=dx\,dy\,d\theta$ on
$\mathcal{M}$ has the suspension form: $dm={\rm Cst}\cdot p\cdot
d\mu_{0}\,dt$. Clearly the $t$-direction is flow-invariant. So we have

$\frac{dT^{-1}_{\mathbf{F}}\mu}{d\mu}(\mathbf{x})=\frac{p(\mathbf{x})}{p(T_{\mathbf{F}}\mathbf{x})}\exp(\int_{0}^{\tau_{\mathbf{F}}(\mathbf{x})}\text{div
}X(\Phi^{t}_{\mathbf{F}}\mathbf{x})\,dt)=\exp(\int_{0}^{\tau_{\mathbf{F}}(\mathbf{x})}ph_{\theta}dt).$
(3.3)

This finishes the proof of the lemma. ∎

##  4\. Properties of $T_{\mathbf{P}}=T_{(\mathbf{F},\mathbf{G})}$

We consider the twisting step of a force pair
$\mathbf{P}=(\mathbf{F},\mathbf{G})$ satisfying the assumptions (A1)–(A4) with
$\|\mathbf{F}\|_{\mathcal{C}^{1}}<\varepsilon$ and
$\|\mathbf{G}-\mathrm{Id}_{M}\|_{\mathcal{C}^{1}}<\varepsilon$. Our main
approach is to compare the combined effect
$T_{\mathbf{P}}\mathbf{x}={\mathbf{G}}\circ T_{\mathbf{F}}\mathbf{x}$ with the
pre-twist map $T_{\mathbf{F}}$.

In Lemma 3.1 we gave a characterization for the forced flow
$\Phi_{\mathbf{F}}$ to be time-reversible. To ensure that $\Phi_{\mathbf{P}}$
is time-reversible, it suffices to know if the twisting process is also time-
reversible. More precisely, let $S=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$,
and $G=\begin{pmatrix}{\bf 0}&S\\\ S&{\bf 0}\end{pmatrix}$, where ${\bf 0}$ is
the $2\times 2$ zero matrix. Then we have the following result.

######  Lemma 4.1.

Assume the condition (3.1) hold. Then $\Phi_{\mathbf{P}}$ is time-reversible
if and only if the graph of ${\mathbf{G}}$ is $G$-invariant.

###### Proof.

Let $T_{\mathbf{P}}\mathbf{x}={\mathbf{G}}\circ T_{\mathbf{F}}\mathbf{x}$ be
decomposition of the forced collision map, $\Phi_{\mathbf{P}}^{-}$ be the
time-reversal flow of $\Phi_{\mathbf{P}}$, and $T_{\mathbf{P}}^{-}$ be the
induced Poincare map of $\Phi_{\mathbf{P}}^{-}$. Let
$\mathcal{I}:\mathcal{M}\to\mathcal{M}$ be the involution on $\mathcal{M}$ and
$I:M\to M$ be the induced involution on $M$.

Under the assumption (3.1), we have that $T^{-}_{\mathbf{F}}=T_{\mathbf{F}}$
and

$T_{\mathbf{P}}^{-}\mathbf{x}=I\circ{\mathbf{G}}^{-1}\circ
T^{-1}_{\mathbf{F}}\circ I(\mathbf{x})=I\circ{\mathbf{G}}^{-1}\circ I\circ
I\circ T^{-1}_{\mathbf{F}}\circ I(\mathbf{x})={\mathbf{G}}^{-}\circ
T_{\mathbf{F}}^{-}(\mathbf{x})={\mathbf{G}}^{-}(T_{\mathbf{F}}\mathbf{x}),$

where ${\mathbf{G}}^{-}=I\circ{\mathbf{G}}^{-1}\circ I$. In order to have
$T_{\mathbf{P}}^{-}=T_{\mathbf{P}}$, a sufficient condition is
${\mathbf{G}}^{-}={\mathbf{G}}$.

Let $(\bar{r},\bar{s})={\mathbf{G}}(r,s)$. Then
${\mathbf{G}}^{-}(\bar{r},-\bar{s})=I\circ{\mathbf{G}}^{-1}(\bar{r},\bar{s})=I(r,s)=(r,-s)$.
So ${\mathbf{G}}^{-}={\mathbf{G}}$ is equivalent to
${\mathbf{G}}(\bar{r},-\bar{s})=(r,-s)$. Combining with the assumption
$(\bar{r},\bar{s})={\mathbf{G}}(r,s)$, we see that
${\mathbf{G}}^{-}={\mathbf{G}}$ is equivalent to the graph of ${\mathbf{G}}$
being $G$-invariant. ∎

######  Remark 3.

In above lemma we take the Poincaré map $T^{-}_{\mathbf{P}}$ of the time-
reversed flow, which can be viewed as the physically time-reversal of the map
$T_{\mathbf{P}}$. We can also define a formally time-reversed map of the
forced map $T_{\mathbf{P}}$ by $\widehat{T}^{-}_{\mathbf{P}}:=I\circ
T^{-1}_{\mathbf{P}}\circ I$. Generally speaking, this formally reversed map
$\widehat{T}^{-}_{\mathbf{P}}$ may not coincide with the Poincaré map
$T^{-}_{\mathbf{P}}$ of the time-reversed flow $\Phi^{-}_{\mathbf{P}}$ (after
we apply a twist force right after the elastic collision). In this paper we
always use the physically time-reversal definition.

By Assumption (A3), the twist force $\mathbf{G}$ preserves tangential
collisions. Therefore, the discontinuity set of $T_{\mathbf{P}}$ is the same
as that of $T_{\mathbf{F}}$, which comprises the preimage of
$\mathcal{S}_{0}:=\\{s=\pm\pi/2\\}$. Similarly, the singularity sets of
$T_{\mathbf{P}}^{-1}$ and $T_{\mathbf{F}}^{-1}$ are the same due to (A3). But
the singular sets for higher iterates are no longer the same. Let
$\mathcal{S}_{\pm n}^{\mathbf{P}}=\cup_{i=0}^{n}T_{\mathbf{P}}^{\mp
i}\mathcal{S}_{0,H}$ with $n\in\mathbb{N}$. Then $T_{\mathbf{P}}^{\pm n}$ is
smooth on all the cells of $M\setminus\mathcal{S}_{\pm n}^{\mathbf{P}}$.

For any phase point $\mathbf{x}=(r,s)\in M$, let
$T_{\mathbf{F}}\mathbf{x}=(r_{1},s_{1})$ and
$T_{\mathbf{P}}\mathbf{x}=(\bar{r}_{1},\bar{s}_{1})$. According to the
discussion between (A2) and (A3), we express the twist force $\mathbf{G}$ in
local coordinates via two smooth functions $g^{1}$ and $g^{2}$ such that

$(\bar{r}_{1},\bar{s}_{1})=\mathbf{G}(r_{1},s_{1})=(r_{1},s_{1})+(g^{1}(r_{1},s_{1}),g^{2}(r_{1},s_{1})).$
(4.1)

Note that $g^{i}$ is a $C^{2}$ function whose $C^{1}$ norm is uniformly
bounded from above by $c\cdot\varepsilon$, for some uniform constant $c>0$.
Moreover $g^{i}(r,\pm 1)=0$, $i=1,2$.

According to (4.1), the differential of the twisting force $\mathbf{G}$
satisfies

$\begin{cases}d\bar{r}_{1}=\left(1+g^{1}_{1}(r_{1},s_{1})\right)dr_{1}+g_{2}^{1}(r_{1},s_{1})ds_{1},\\\
d\bar{s}_{1}=g^{2}_{1}(r_{1},s_{1})dr_{1}+\left(1+g_{2}^{2}(r_{1},s_{1})\right)ds_{1},\end{cases}$
(4.2)

where $g^{i}_{1}=\partial g^{i}/\partial r$ and $g^{i}_{2}=\partial
g^{i}/\partial s$. So the differential of the map $T_{\mathbf{P}}$ is given by

$\displaystyle D_{\mathbf{x}}T_{\mathbf{P}}$
$\displaystyle=D_{T_{\mathbf{F}}\mathbf{x}}\mathbf{G}\circ
D_{\mathbf{x}}T_{\mathbf{F}}.$ (4.3)

Note that $T_{\mathbf{P}}$ may not be a $C^{1}$-perturbation of
$T_{\mathbf{F}}$, since $T_{\mathbf{F}}$ is unbounded around the boundary of
$M$. However, it follows from Eq. (4.3) that

$\det
D_{\mathbf{x}}T_{\mathbf{P}}=\mathcal{J}_{\mathbf{G}}(r_{1},s_{1})\cdot\det
D_{\mathbf{x}}T_{\mathbf{F}}=\left(1+\hat{g}(r_{1},s_{1})\right)\cdot\det
D_{\mathbf{x}}T_{\mathbf{F}},$ (4.4)

where

$\displaystyle\hat{g}(r_{1},s_{1})$
$\displaystyle=\mathcal{J}_{\mathbf{G}}(r_{1},s_{1})-1$
$\displaystyle=g^{1}_{1}(r_{1},s_{1})+g^{2}_{2}(r_{1},s_{1})+g^{1}_{1}(r_{1},s_{1})g^{2}_{2}(r_{1},s_{1})-g^{1}_{2}(r_{1},s_{1})g^{2}_{1}(r_{1},s_{1}).$
(4.5)

Note that $\mathcal{J}_{\mathbf{G}}$ is a $C^{1}$ function with
$\hat{g}(r_{1},s_{1})=\mathcal{O}(\varepsilon)$. So we have that

$\displaystyle\det
D_{\mathbf{x}}T_{\mathbf{P}}=\left(1+\hat{g}(r_{1},s_{1})\right)\cdot\exp(\int_{0}^{\tau_{\mathbf{F}}(\mathbf{x})}ph_{\theta}dt)=1+\mathcal{O}(\varepsilon).$
(4.6)

Once again, let $\mu_{0}={\rm Cst}\,dr\,ds$ be the normalized Lebesgue measure
on $M$,
$\mathcal{J}_{\mathbf{P}}(\mathbf{x}):=dT_{\mathbf{P}}^{-1}\mu_{0}/d\mu_{0}(\mathbf{x})$
be the density function defined by $T_{\mathbf{P}}$. Note that
$\mathcal{J}_{\mathbf{P}}(\mathbf{x})=\det D_{\mathbf{x}}T_{\mathbf{P}}$. So
we have:

######  Lemma 4.2.

Let $\mathbf{P}=(\mathbf{F},\mathbf{G})$ be an $\varepsilon$-small force pair,
and $T_{\mathbf{P}}$ the forced collision map. Let $\mathbf{x}=(r,s)\in M$,
$T_{\mathbf{F}}\mathbf{x}=(r_{1},s_{1})$ and
$T_{\mathbf{P}}\mathbf{x}=(\bar{r}_{1},\bar{s}_{1})$. Then the Jacobian
$\mathcal{J}_{\mathbf{P}}(\mathbf{x})$ satisfies

$1-\mathcal{J}_{\mathbf{P}}(\mathbf{x})=\varepsilon
H(\mathbf{x})+\varepsilon^{2}R_{\mathbf{P}},$ (4.7)

where

$\displaystyle H(r,s)$
$\displaystyle=\frac{1}{\varepsilon}\left(2-\exp(\int_{0}^{\tau_{\mathbf{F}}(r,s)}ph_{\theta}dt)-\mathcal{J}_{\mathbf{G}}(T_{\mathbf{F}}\mathbf{x})\right)$
$\displaystyle=\frac{1}{\varepsilon}\left(1-\exp(\int_{0}^{\tau_{\mathbf{F}}(r,s)}ph_{\theta}dt)-\hat{g}(r_{1},s_{1})\right).$

Moreover, $\mu_{0}(H)=0$, and both $H$ and $R_{\mathbf{P}}$ are uniformly
bounded and $C^{1}$ on each component of $M\backslash\mathcal{S}^{T}_{1}$.

###### Proof.

According to Chain Rule and Lemma 3.2, we have

$\displaystyle\mathcal{J}_{\mathbf{P}}(\mathbf{x})$
$\displaystyle=\frac{dT^{-1}_{\mathbf{P}}\mu_{0}}{d\mu_{0}}(\mathbf{x})=\frac{dT^{-1}_{\mathbf{P}}\mu_{0}}{dT^{-1}_{\mathbf{F}}\mu_{0}}(\mathbf{x})\cdot\frac{dT^{-1}_{\mathbf{F}}\mu_{0}}{d\mu_{0}}(\mathbf{x})$
$\displaystyle=\mathcal{J}_{\mathbf{G}}(T_{\mathbf{F}}\mathbf{x})\cdot\det
D_{\mathbf{x}}T_{\mathbf{F}}=\left(1+\hat{g}(r_{1},s_{1})\right)\cdot\exp(\int_{0}^{\tau_{\mathbf{F}}(\mathbf{x})}ph_{\theta}dt).$
(4.8)

Let
$H(\mathbf{x})=(1-\exp(\int_{0}^{\tau_{\mathbf{F}}(r,s)}ph_{\theta}dt)-\hat{g}(r_{1},s_{1}))/\varepsilon$.
Then we get the following expansion for $\mathcal{J}_{\mathbf{P}}$ at
$\mathbf{P}=0$:

$\mathcal{J}_{\mathbf{P}}(\mathbf{x})=\mathcal{J}_{0}(\mathbf{x})-\varepsilon
H(\mathbf{x})-\varepsilon^{2}R_{\mathbf{P}},$ (4.9)

where $R_{\mathbf{P}}$ is the residual term (up to a factor $\epsilon^{2}$).

Now it is easy to see that $\mu_{0}(H)=0$. Firstly,
$1+\mu_{0}(\hat{g})=\mu_{0}(1+\hat{g})=\mu_{0}(\mathcal{J}_{\mathbf{G}})=\mu_{0}(M)=1$.
So $\mu_{0}(\hat{g})=0$. Secondly, we note that
$\mu_{0}(e^{\int_{0}^{\tau_{\mathbf{F}}(r,s)}ph_{\theta}dt})=\mu_{0}(DT_{\mathbf{F}})=\mu_{0}(T_{\mathbf{F}}M)=\mu_{0}(M)=1$.
Therefore, we have $\mu_{0}(H)=0$.

By assumptions (A1)–(A3), both $H$ and $R_{\mathbf{P}}$ are uniformly bounded
on $M$, and are $C^{1}$ continuous functions on each component of
$M\backslash\mathcal{S}^{\mathbf{P}}_{1}$. This completes the proof. ∎

Let $\mu_{\mathbf{P}}$ be the SRB measure of $T_{\mathbf{P}}$ on $M$ given by
Theorem 2.1. This measure represents the natural non-equilibrium steady state
(NESS) for the system (see [Ru99]), which might be singular with respect to
the Liouville measure on $M$. There are several physically interested
quantities associated to the NESS $\mu_{\mathbf{P}}$. For example, the current
of the billiard flow on the $\mathbb{Z}^{2}$-periodic table $\widetilde{Q}$ is
given by

$\mathbf{J}_{\mathbf{P}}=\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}})=\mu^{o}_{\mathbf{P}}(\widetilde{\boldsymbol{\Delta}}_{\mathbf{P}}),$

where ${\boldsymbol{\Delta}}_{\mathbf{P}}$ is the induced displacement vector
function on $M$, $\mu_{\mathbf{P}}^{o}$ is a copy of $\mu_{\mathbf{P}}$ on a
fundamental domain (say $\widetilde{M}_{o}$) of $\widetilde{M}$. This current
was derived in [CELS93a, CELS93b] when the billiard is moving in an electric
field with a Gaussian thermostat. The study of the current as a function of a
general electric field was carried out in [Ch01, Ch08]. In the other
direction, these results were generalized in [Zh11] to systems where the
collision rule is perturbed. Here we study the current for billiards under
more general external forces. Firstly we will prove the linear response
property for general observable with respect to the perturbed billiard system.

######  Lemma 4.3.

Let $\mathbf{P}=(\mathbf{F},\mathbf{G})$ be an $\varepsilon$-small force pair,
$T_{\mathbf{P}}$ be the induced billiard map and $\mu_{\mathbf{P}}$ be the SRB
measure of $T_{\mathbf{P}}$ on $M$, $\mathcal{H}$ be the set of piecewise
Holder continuous functions on $M$ whose discontinuities occur at the
singularities $S_{1}^{\mathbf{P}}$. Then for any $f\in\mathcal{H}$, we have

$\mu_{\mathbf{P}}(f)=\mu_{0}(f)+\varepsilon\cdot\sum_{k=1}^{\infty}\mu_{0}[(f\circ
T^{k}_{0})\cdot H]+o(\varepsilon)$ (4.10)

###### Proof.

Let $f\in\mathcal{H}$. Then for all $n\geq 1$, we have the following identity:

$\displaystyle T_{\mathbf{P}}^{n}\mu_{0}(f)-\mu_{0}(f)$
$\displaystyle=\sum_{k=1}^{n}(T_{\mathbf{P}}^{k}\mu_{0}(f)-T_{\mathbf{P}}^{k-1}\mu_{0}(f))=\sum_{k=1}^{n}\mu_{0}[(f\circ
T^{k}_{\mathbf{P}})(1-\mathcal{J}_{\mathbf{P}})]$
$\displaystyle=\sum_{k=1}^{n}\mu_{0}[(f\circ T^{k}_{\mathbf{P}})(\varepsilon
H(\mathbf{x})+\varepsilon^{2}R_{\mathbf{P}})].$

Note that $T_{\mathbf{P}}^{n}\mu_{0}(f)\to\mu_{\mathbf{P}}(f)$ exponentially
(by Theorem 2.1-a). Passing $n\to\infty$, we get that

$\displaystyle\mu_{\mathbf{P}}(f)$
$\displaystyle=\lim_{n\to\infty}T_{\mathbf{P}}^{n}\mu_{0}(f)=\mu_{0}(f)+\sum_{k=1}^{\infty}\mu_{0}[(f\circ
T^{k}_{\mathbf{P}})(\varepsilon
H(\mathbf{x})+\varepsilon^{2}R_{\mathbf{P}})].$ (4.11)

It follows from the fact that $\mathcal{J}_{\mathbf{P}}$ is the density
function of a probability measure, that
$\mu_{0}(1-\mathcal{J}_{\mathbf{P}})=0$. Combining with the fact that
$\mu_{0}(H)=0$, we get that $\mu_{0}(R_{\mathbf{P}})=0$, too. In addition $H$
and $R_{\mathbf{P}}$ are piecewise $C^{2}$ functions whose discontinuities
occur only at the singularities of $T_{\mathbf{P}}$. Thus $H$ and
$R_{\mathbf{P}}$ belong to $\mathcal{H}$. Now (4.11) implies that

$\displaystyle\sum_{k=1}^{\infty}\mu_{0}[(f\circ
T^{k}_{\mathbf{P}})(\varepsilon
H+\varepsilon^{2}R_{\mathbf{P}})]=\varepsilon\sum_{k=1}^{\infty}\mu_{0}[(f\circ
T^{k}_{\mathbf{P}})H]+\varepsilon^{2}\sum_{k=1}^{\infty}\mu_{0}[(f\circ
T^{k}_{\mathbf{P}})R_{\mathbf{P}}]$ $\displaystyle=$
$\displaystyle\varepsilon\sum_{k=1}^{\infty}\mu_{0}[(f\circ
T^{k}_{0})H]+\varepsilon\sum_{k=1}^{\infty}\mu_{0}[(f\circ
T^{k}_{\mathbf{P}}-f\circ
T^{k}_{0})H]+\varepsilon^{2}\sum_{k=1}^{\infty}\mu_{0}[(f\circ
T^{k}_{\mathbf{P}})R_{\mathbf{P}}]$ $\displaystyle=$
$\displaystyle\varepsilon\sum_{k=1}^{\infty}\mu_{0}[(f\circ
T^{k}_{0})H]+o(\varepsilon),$ (4.12)

where we have used the Lebesgue Dominated Convergence Theorem in the last
step, since both $\sum_{k=1}^{\infty}\mu_{0}[(f\circ T^{k}_{\mathbf{P}})H]$
and $\sum_{k=1}^{\infty}\mu_{0}[(f\circ T^{k}_{0})H]$ converge exponentially
fast (by Theorem 2.1-b) and $T^{k}_{\mathbf{P}}\to T^{k}_{0}$ as
$\mathbf{P}\to 0$. Therefore the series are dominated by $o(1)$. This
completes the proof. ∎

Let $(x,y)$ be the coordinates of the position of the particle
$\tilde{\mathbf{q}}\in\widetilde{Q}$, and

$\widetilde{\boldsymbol{\Delta}}_{\mathbf{P}}:=(\Delta_{x,\mathbf{P}},\Delta_{y,\mathbf{P}})=\tilde{\pi}\circ\widetilde{T}_{\mathbf{P}}-\tilde{\pi},$
(4.13)

be the displacement vector map defined on $\widetilde{M}$. As pointed out
before, $\widetilde{\boldsymbol{\Delta}}_{\mathbf{P}}$ is
$\mathbb{Z}^{2}$-periodic and hence induces a well-defined,
$\mathbb{R}^{2}$-valued function ${\boldsymbol{\Delta}}_{\mathbf{P}}$ on $M$.
Using the fact that $\mu_{0}({\boldsymbol{\Delta}}_{0})=0$ and
$\mu_{0}(\tau_{0})=\bar{\tau}$, we get the following results.

######  Lemma 4.4.

Both functions ${\boldsymbol{\Delta}}_{\mathbf{P}}$ and $\tau_{\mathbf{P}}$
belong to $\mathcal{H}$ with Holder exponent $\frac{1}{2}$ and uniformly
bounded. Moreover,

$\mu_{0}(\Delta_{\mathbf{P}})=\mathcal{O}({\varepsilon})\,\,\,\,\,\text{ and
}\,\,\,\,\,\mu_{0}(\tau_{\mathbf{P}})=\bar{\tau}+\mathcal{O}({\varepsilon})$

The proof is omitted here, since Hölder continuity follows closely from [DC09,
Lemma 8.2] and the estimations are almost identical with the proof of Lemma
8.1 in [Zh11].

Next we calculate the current for the moving billiards under small force
depending on $\varepsilon$. We only consider the case when the new system
$T_{\mathbf{P}}$ is time-reversible.

######  Theorem 4.5.

Suppose the system is time-reversible under the external forces
$\mathbf{P}=(\mathbf{F},\mathbf{G})$. Then

$\displaystyle\mathbf{J}_{\mathbf{P}}:=\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}})=\varepsilon{\boldsymbol{\sigma}}+o(\varepsilon),$
(4.14)

where ${\boldsymbol{\sigma}}=(\sigma_{x},\sigma_{y})$ is given by

$\sigma_{a}=\frac{1}{2}\mu_{0}(\Delta_{a,\mathbf{P}}\cdot
H)+\sum_{k=1}^{\infty}\mu_{0}[(\Delta_{a,\mathbf{P}}\circ T^{k}_{0})\cdot
H],\quad a\in\\{x,y\\}.$ (4.15)

Moreover, the current for the flow satisfies
$\hat{\mathbf{J}}_{\mathbf{P}}=\varepsilon{\boldsymbol{\sigma}}/\bar{\tau}_{\mathbf{P}}+o(\varepsilon)$,
where $\bar{\tau}=\mu_{0}(\tau_{0})$.

###### Proof.

We first use the invariance of $\mu_{\mathbf{P}}$ and the Kawasaki formula
(4.11) to write $\mathbf{J}_{\mathbf{P}}$ as

$\displaystyle\mathbf{J}_{\mathbf{P}}=\mu_{\mathbf{P}}(\Delta_{\mathbf{P}})=\frac{1}{2}(\mu_{\mathbf{P}}(\Delta_{\mathbf{P}})+\mu_{\mathbf{P}}(\Delta_{\mathbf{P}}\circ
T^{-1}_{\mathbf{P}}))$ (4.16) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\mu_{0}(\Delta_{\mathbf{P}})+\mu_{0}(\Delta_{\mathbf{P}}\circ
T^{-1}_{\mathbf{P}})+\mu_{0}[\Delta_{\mathbf{P}}(1-\mathcal{J}_{\mathbf{P}})]\right)+\sum_{k=1}^{\infty}\mu_{0}[(\Delta_{\mathbf{P}}\circ
T^{k}_{\mathbf{P}})(1-\mathcal{J}_{\mathbf{P}})].$

Recall that $\widetilde{M}$ is the collision space of the
$\mathbb{Z}^{2}$-periodic table $\widetilde{Q}\subset\mathbb{R}^{2}$. Let
$\widetilde{M}_{o}$ be a fundamental domain of $\widetilde{M}$, say
$(\partial\widetilde{Q}\cap[0,1]^{2})\times[-1,1]$, and $\mu_{0}^{o}={\rm
Cst}\,dr\,ds$ be a copy of $\mu_{0}$ on $\widetilde{M}_{o}$. So for any
function $f$ on $M$, $\mu_{0}(f)=\mu_{0}^{o}(\tilde{f})$, where $\tilde{f}$ is
a $\mathbb{Z}^{2}$-periodic lift of $f$ on $\widetilde{M}$.

Since $\mu_{0}^{o}$ is evenly distributed on $\widetilde{M}_{o}$, it is
invariant under the involution $I$. In particular, we have

$\mu_{0}^{o}(\tilde{\pi}\circ\widetilde{T}_{\mathbf{P}})=\mu_{0}^{o}(\tilde{\pi}\circ\widetilde{T}^{-}_{\mathbf{P}})=\mu_{0}^{o}(\tilde{\pi}\circ
I\circ\widetilde{T}^{-1}_{\mathbf{P}}\circ
I)=\mu_{0}^{o}(\tilde{\pi}\circ\widetilde{T}_{\mathbf{P}}^{-1}).$ (4.17)

This implies that

$\displaystyle\mu_{0}({\boldsymbol{\Delta}}_{\mathbf{P}})+\mu_{0}({\boldsymbol{\Delta}}_{\mathbf{P}}\circ
T^{-1}_{\mathbf{P}})=\mu_{0}^{o}(\widetilde{\boldsymbol{\Delta}}_{\mathbf{P}})+\mu_{0}^{o}(\widetilde{\boldsymbol{\Delta}}_{\mathbf{P}}\circ\widetilde{T}^{-1}_{\mathbf{P}})$
$\displaystyle=$
$\displaystyle\mu_{0}^{o}(\tilde{\pi}\circ\widetilde{T}_{\mathbf{P}}-\tilde{\pi})+\mu_{0}^{o}(\tilde{\pi}-\tilde{\pi}\circ\widetilde{T}^{-1}_{\mathbf{P}})=\mu_{0}^{o}(\tilde{\pi}\circ\widetilde{T}_{\mathbf{P}})-\mu_{0}^{o}(\tilde{\pi}\circ\widetilde{T}_{\mathbf{P}}^{-1})=0.$

See [CELS93a, DC09] for related discussions. Then according to (4.7), the two
components of the current are given by

$\mu_{\mathbf{P}}(\Delta_{a,\mathbf{P}})=\frac{1}{2}\mu_{0}[\Delta_{a,\mathbf{P}}(\varepsilon
H+\varepsilon^{2}\mathcal{R})]+\sum_{k=1}^{\infty}\mu_{0}[(\Delta_{a,\mathbf{P}}\circ
T^{k}_{\mathbf{P}})\cdot(\varepsilon
H+\varepsilon^{2}\mathcal{R})]=\varepsilon\sigma_{a}+o(\varepsilon),$

where

$\displaystyle\sigma_{a}=\frac{1}{2}\mu_{0}(\Delta_{a,\mathbf{P}}\cdot
H)+\sum_{k=1}^{\infty}\mu_{0}[(\Delta_{a,\mathbf{P}}\circ T^{k}_{0})\cdot H].$
(4.18)

Here we use a similar argument in the last step as in (4), since
$T_{\mathbf{P}}\to T_{0}$ as $\mathbf{P}\to 0$.

Denote ${\boldsymbol{\sigma}}=(\sigma_{x},\sigma_{y})$. So we have shown that
$\mathbf{J}_{\mathbf{P}}=\varepsilon{\boldsymbol{\sigma}}+o(\varepsilon)$. The
current of the forced flow $\Phi_{\mathbf{P}}$ is given by
$\hat{\mathbf{J}}_{\mathbf{P}}=\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}})/\bar{\tau}_{\mathbf{P}}=\varepsilon{\boldsymbol{\sigma}}/\bar{\tau}_{\mathbf{P}}+o(\varepsilon)$,
where $\bar{\tau}_{\mathbf{P}}=\mu_{\mathbf{P}}(\tau_{\mathbf{P}})$. This
completes the proof. ∎

######  Remark 4.

It is worth to point out that $\mu_{0}(H)=0$ is crucial to define
$\sigma_{a}$, $a\in\\{x,y\\}$. Otherwise the series in (4.18) used to define
$\sigma_{a}$ don’t even converge if we take the linear term
$-\int_{0}^{\tau_{\mathbf{F}}(r,s)}ph_{\theta}dt$ . This is the main reason
why we keep all nonlinear terms in Lemma 4.2. However, we can use the linear
term $\int_{0}^{\tau_{\mathbf{F}}(r,s)}ph_{\theta}dt$ in all physical models
with a Gaussian thermostat (see [MHB87, CELS93a, CELS93b, Ch01]). This is
exactly the content in Proposition 2.6, 2.7 and Corollary 2.8.

###### Proof of Proposition 2.6.

When restricted to the level set $\mathcal{E}_{1}=\\{\|\mathbf{p}\|\equiv
1\\}$, we have $h=F$. Then we only need to prove that the series (4.18) do
converge with the linear choice
$H=\int_{0}^{\tau_{\mathbf{F}}(\mathbf{x})}F_{\theta}\,dt$. By Theorem 2.1-b
and Lemma 4.4, it is sufficient to show that
$\mu_{0}(\int_{0}^{\tau_{\mathbf{F}}(\mathbf{x})}F_{\theta}\,dt)=0$. Then
according to the fact that $dm={\rm Cst}\cdot p_{0}\cdot
d\mu_{0}\,dt=dx\,dy\,d\theta$, we get

$\displaystyle\mu_{0}(\int_{0}^{\tau_{\mathbf{E}}(\mathbf{x})}F_{\theta}\,dt)=$
$\displaystyle{\rm
Cst}\cdot\int_{M}\int_{0}^{\tau_{\mathbf{E}}(\mathbf{x})}F_{\theta}\,dtd\mu_{0}=\int_{\mathcal{M}}F_{\theta}dm$
$\displaystyle=$
$\displaystyle\int_{Q}\left(\int_{S^{1}}F_{\theta}\,d\theta\right)dx\,dy=\int_{Q}\left(F|_{\partial
S^{1}}\right)dx\,dy=0.$ (4.19)

This finishes the proof of Proposition 2.6. ∎

###### Proof of Proposition 2.7.

Let $\mathbf{E}=(\varepsilon e_{1}(\mathbf{q}),\varepsilon e_{2}(\mathbf{q}))$
be the electric field on $Q$, $\dot{\mathbf{q}}=\mathbf{p}$,
$\dot{\mathbf{p}}=\mathbf{E}-\alpha\mathbf{p}$ be the thermostatted system.
When restricted to the level set $\mathcal{E}_{1}=\\{\|\mathbf{p}\|\equiv
1\\}$, the function $h=-e_{1}\sin\theta+e_{2}\cos\theta$, and
$h_{\theta}=-e_{1}\cos\theta-e_{2}\sin\theta=-(e_{1},e_{2})\cdot\mathbf{p}$.
So
$H=-\int_{0}^{\tau_{\mathbf{F}}(\mathbf{x})}h_{\theta}\,dt=(e_{1},e_{2})\cdot\int_{0}^{\tau_{\mathbf{E}}(\mathbf{x})}\mathbf{p}\,dt=(e_{1},e_{2})\cdot{\boldsymbol{\Delta}}_{\mathbf{E}}(\mathbf{x})$.
This finishes the proof of Proposition 2.7. ∎

######  Remark 5.

There is another classical representation of the current of the billiard flow
$\hat{\mathbf{J}}=\hat{\mu}(\mathbf{p})$, which may not hold for general
forces. More precisely, let $\hat{\mu}_{\mathbf{P}}$ be the SRB measure of the
flow $\Phi_{\mathbf{P}}$, which can also be obtained as the suspension of
$\mu_{\mathbf{P}}$ on $\mathcal{M}$. Then
$\hat{\mathbf{J}}_{\mathbf{P}}=\hat{\mu}_{\mathbf{P}}(\mathbf{p})$ holds if
and only if
$\displaystyle{\boldsymbol{\Delta}}_{\mathbf{P}}(\mathbf{x})={\boldsymbol{\Delta}}_{\mathbf{F}}(\mathbf{x}):=\int_{0}^{\tau_{\mathbf{F}}(\mathbf{x})}\mathbf{p}dt$,
that is, there is no slip on $\partial Q$ after each collision. For a general
twist force with slip, we would have an additional term
$\displaystyle{\boldsymbol{\Delta}}_{\mathbf{G}}(T_{\mathbf{F}}\mathbf{x})={\boldsymbol{\Delta}}_{\mathbf{P}}(\mathbf{x})-{\boldsymbol{\Delta}}_{\mathbf{F}}(\mathbf{x})$.
So we have
$\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}})=\int_{M_{o}}(\tilde{\pi}(\widetilde{T}_{\mathbf{P}}\mathbf{x})-\tilde{\pi}(\widetilde{T}_{\mathbf{F}}\mathbf{x}))d\mu_{\mathbf{P}}^{o}+\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{F}})$.
Since $\mu_{\mathbf{P}}$ is $T_{\mathbf{P}}$-invariant, we get an modified
formula of the current for the forced flow

$\hat{\mathbf{J}}_{\mathbf{P}}=\hat{\mu}_{\mathbf{P}}(\mathbf{p})+\frac{1}{\mu_{\mathbf{P}}(\tau_{\mathbf{P}})}\int_{M_{o}}(\tilde{\pi}(\mathbf{x})-\tilde{\pi}({\mathbf{G}}^{-1}\mathbf{x}))d\mu^{o}_{\mathbf{P}}.$
(4.20)

Now we turn to the proof of (2.9) and (2.11).

###### Proof of Theorem 2.2 and Theorem 2.3.

The convergence of of the distribution
$\frac{\tilde{\mathbf{q}}_{n}-n\mathbf{J}_{\mathbf{P}}}{\sqrt{n}}$ to a normal
law $\mathcal{N}(\textbf{0},\mathbf{D}_{\mathbf{P}})$ follows directly from
the central limit theorem (see Theorem 2.1-c), since
$\tilde{\mathbf{q}}_{n}=\sum_{k=0}^{n-1}{\boldsymbol{\Delta}}_{\mathbf{P}}\circ
T^{k}_{\mathbf{P}}$, where ${\boldsymbol{\Delta}}_{\mathbf{P}}$ belongs to
$\mathcal{H}$ by Lemma 4.4. Thus it is enough to estimate the covariance
matrix $\mathbf{D}_{\mathbf{P}}$, which is given by the following sum of
correlations

$\mathbf{D}_{\mathbf{P}}=\sum_{n=-\infty}^{\infty}\left[\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}}\circ
T^{n}_{\mathbf{P}}\otimes{\boldsymbol{\Delta}}_{\mathbf{P}})-\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}})\otimes\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}})\right].$
(4.21)

Note that for any $n\geq 1$,

$\lim_{\mathbf{P}\to 0}{\boldsymbol{\Delta}}_{\mathbf{P}}\circ
T^{n}_{\mathbf{P}}={\boldsymbol{\Delta}}_{0}\circ T^{n}_{0}.$

Furthermore by Theorem 2.1, there exist $C>0$ and $\theta\in(0,1)$, such that
for any $\varepsilon$-small force pair $\mathbf{P}$ and for all
$a,b\in\\{x,y\\}$,

$|\mathcal{C}_{a,b}(n)|=|\mu_{\mathbf{P}}(\Delta_{a,\mathbf{P}}\circ
T^{n}_{\mathbf{P}}\cdot\Delta_{b,\mathbf{P}})-\mu_{\mathbf{P}}(\Delta_{a,\mathbf{P}})\cdot\mu_{\mathbf{P}}(\Delta_{b,\mathbf{P}})|\leq
C\theta^{|n|}.$

Therefore the series $\sum_{n=-\infty}^{\infty}\mathcal{C}_{a,b}(n)$ converges
uniformly (and exponentially) for all $a,b\in\\{x,y\\}$. So the diffusion
matrix varies continuously at $\mathbf{P}=0$:

$\displaystyle\mathbf{D}_{\mathbf{P}}$
$\displaystyle=\sum_{n=-\infty}^{\infty}[\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}}\circ
T^{n}_{\mathbf{P}}\otimes{\boldsymbol{\Delta}}_{\mathbf{P}})-\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}})\otimes\mu_{\mathbf{P}}({\boldsymbol{\Delta}}_{\mathbf{P}})]$
$\displaystyle=\sum_{n=-\infty}^{\infty}\mu_{0}({\boldsymbol{\Delta}}_{0}\circ
T^{n}_{0}\otimes{\boldsymbol{\Delta}}_{0})+o(1)=\mathbf{D}_{0}+o(1),$ (4.22)

since $\mu_{0}({\boldsymbol{\Delta}}_{0})=0$. This finishes the proof of
Theorem 2.2. Theorem 2.3 follows directly from Theorem 2.2. ∎

Finally we prove Theorem 2.4.

###### Proof of Theorem 2.4.

Let $\lambda_{\mathbf{P}}^{s}<0<\lambda^{u}_{\mathbf{P}}$ denote the Lyapunov
exponents of the ergodic system $(T_{\mathbf{P}},\mu_{\mathbf{P}})$. The sum
$\xi_{\mathbf{P}}:=-(\lambda_{\mathbf{P}}^{s}+\lambda^{u}_{\mathbf{P}})$
represents the physical entropy production rate of the perturbed system
$(T_{\mathbf{P}},\mu_{\mathbf{P}})$. Then by Oseledets Multiplicity Ergodic
Theorem, we have

$\xi_{\mathbf{P}}=-(\lambda_{\mathbf{P}}^{s}+\lambda_{\mathbf{P}}^{u})=-\mu_{\mathbf{P}}(\log\mathcal{J}_{\mathbf{P}}).$

Note that $1-\mathcal{J}_{\mathbf{P}}=\mathcal{O}(\epsilon)$. So we have

$\displaystyle\xi_{\mathbf{P}}$
$\displaystyle=-\mu_{\mathbf{P}}(\log(1-(1-\mathcal{J}_{\mathbf{P}})))=\mu_{\mathbf{P}}(1-\mathcal{J}_{\mathbf{P}})\frac{1}{2}\mu_{\mathbf{P}}((1-\mathcal{J}_{\mathbf{P}})^{2})+\mathcal{O}(\varepsilon^{3}).$

Using the similar analysis as in (4) and
$\mu_{0}(\mathcal{J}_{\mathbf{P}})=1$, one can check that

$\displaystyle\mu_{\mathbf{P}}(1-\mathcal{J}_{\mathbf{P}})=\mu_{0}(1-\mathcal{J}_{\mathbf{P}})+\sum_{k=1}^{\infty}\mu_{0}[(1-\mathcal{J}_{\mathbf{P}})\circ
T_{\mathbf{P}}^{k}\cdot(1-\mathcal{J}_{\mathbf{P}})]$ $\displaystyle=$
$\displaystyle\varepsilon^{2}\sum_{k=1}^{\infty}\mu_{0}(H\circ
T_{\mathbf{P}}^{k}\cdot
H)+\mathcal{O}(\varepsilon^{3})=\varepsilon^{2}\sum_{k=1}^{\infty}\mu_{0}(H\circ
T_{0}^{k}\cdot H)+o(\varepsilon^{2}).$ (4.23)

In addition, we have

$\displaystyle\mu_{\mathbf{P}}((1-\mathcal{J}_{\mathbf{P}})^{2})$
$\displaystyle=\mu_{0}((1-\mathcal{J}_{\mathbf{P}})^{2})+\sum_{k=1}^{\infty}\mu_{0}[((1-\mathcal{J}_{\mathbf{P}})^{2})\circ
T_{\mathbf{P}}^{k}\cdot(1-\mathcal{J}_{\mathbf{P}})]$
$\displaystyle=\varepsilon^{2}\mu_{0}(H^{2})+\frac{\varepsilon^{3}}{2}\sum_{k=1}^{\infty}\mu_{0}(H^{2}\circ
T_{0}^{k}\cdot H)+o(\varepsilon^{3}),$ (4.24)

where we used (4.7) and (4.11) in the above estimates. Combining these facts,
we get

$\xi_{\mathbf{P}}=\frac{\varepsilon^{2}}{2}\mu_{0}(H^{2})+\varepsilon^{2}\sum_{k=1}^{\infty}\mu_{0}(H\circ
T_{0}^{k}\cdot
H)+o(\varepsilon^{2})=\varepsilon^{2}\cdot\frac{\sigma^{2}_{0}(H)}{2}+o(\varepsilon^{2}),$

where $\sigma^{2}_{0}(H)=\sum_{k=-\infty}^{\infty}\mu_{0}[H\circ
T_{0}^{k}\cdot H]$. Being an SRB measure, the metric entropy of
$\mu_{\mathbf{P}}$ satisfies
$h_{\mu_{\mathbf{P}}}(T_{\mathbf{P}})=\lambda_{\mathbf{P}}^{u}=\mu_{\mathbf{P}}(\Lambda^{u}_{\mathbf{P}})$,
by Pesin’s Entropy Formula and by Birkhoff Ergodic Theorem, respectively,
where $\Lambda^{u}_{\mathbf{P}}(\mathbf{x})$ is the local expansion rate of
$\mathbf{x}\in M$ along the unstable direction under map $T_{\mathbf{P}}$.
Then according to (4.11) and the exponential decay of correlations, we have

$\displaystyle h_{\mu_{\mathbf{P}}}(T_{\mathbf{P}})$
$\displaystyle=\mu_{0}(\Lambda^{u}_{\mathbf{P}})+\sum_{k=1}^{\infty}\mu_{0}(\Lambda^{u}_{\mathbf{P}}\circ
T^{k}_{\mathbf{P}}\cdot(1-\mathcal{J}_{\mathbf{P}}))$
$\displaystyle=\mu_{0}(\Lambda^{u}_{0})+\sum_{k=1}^{\infty}\mu_{0}(\Lambda^{u}_{0}\circ
T_{\mathbf{P}}^{k}\cdot(\varepsilon H+\varepsilon^{2}R_{\mathbf{P}}))+o(1)$
$\displaystyle=h_{0}+o(1),$ (4.25)

since $\Lambda^{u}_{\mathbf{P}}\to\Lambda^{u}_{0}$ as $\mathbf{P}\to 0$. Here
$h_{0}:=h_{\mu_{0}}(T_{0})>0$ is the metric entropy of the unforced billiard
map $T_{0}$. Combining the above facts and Young’s Dimension Formula (2.17) in
[You82], we get

$\displaystyle\mathrm{HD}(\mu_{\mathbf{P}})$
$\displaystyle=h_{\mu_{\mathbf{P}}}(T_{\mathbf{P}})\left(\frac{1}{\lambda^{u}_{\mathbf{P}}}-\frac{1}{\lambda^{s}_{\mathbf{P}}}\right)=1-\frac{\lambda^{u}_{\mathbf{P}}}{\lambda^{s}_{\mathbf{P}}}=2-\frac{\xi_{\mathbf{P}}}{h_{\mu_{\mathbf{P}}}(T_{\mathbf{P}})+\xi_{\mathbf{P}}}$
$\displaystyle=2-\frac{\varepsilon^{2}\cdot\sigma^{2}_{0}(H)/2+o(\varepsilon^{2})}{h_{0}+\varepsilon^{2}\cdot\sigma^{2}_{0}(H)/2+o(1)}$
$\displaystyle=2-\varepsilon^{2}\cdot\frac{\sigma^{2}_{0}(H)}{2h_{0}}+o(\varepsilon^{2}).$

This completes the proof of Theorem 2.4. ∎

Then by the suspension property, we see that the dimension of the measure
$\hat{\mu}_{\mathbf{P}}$ is given by

$\mathrm{HD}(\hat{\mu}_{\mathbf{P}})=\mathrm{HD}(\mu_{\mathbf{P}})+1=3-\varepsilon^{2}\cdot\frac{\sigma^{2}_{H}}{2h_{0}}+o(\varepsilon^{2}).$

So if $\varepsilon$ is small and $\sigma^{2}_{H}>0$, then
$1<\mathrm{HD}(\mu_{\mathbf{P}})<2$ and
$2<\mathrm{HD}(\hat{\mu}_{\mathbf{P}})<3$. Therefore, both $\mu_{\mathbf{P}}$
and $\hat{\mu}_{\mathbf{P}}$ are singular with respect to the Lebesgue
measures and admit the fractal structures if $\sigma^{2}_{H}\neq 0$ under a
small force pair $\mathbf{P}$. In fact, it is believed [CELS93b] (based on
numerical evidences) that $\mu_{bP}$ should be multifractal with a continuous
spectrum of fractal dimensions.

Acknowledgement. N.C. was partially supported by National Science Foundation,
grant DMS-0969187. H.-K.Z. was partially supported by National Science
Foundation, grant DMS-1151762, and by the French CNRS with a poste d’accueil
position at the Center of Theoretical Physics in Luminy. P.Z. was partially
supported by NNSF project (11071231).

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