## 1. Context

## 2. Opposite arrows

## 3. Dialogue on opposite arrows

#### 3.1 Solving the two-time boundary value problem

^{N}(with 2N the phase space dimension). I don’t have a general answer, but can offer an informative example. Appendix A of Sec. 5.0 in [8] presents the following result: even nonequilibrium initial conditions imply a tremendous reduction in available phase space; two-time macroscopic boundary conditions are also a tremendous restriction, but not more serious than slightly more demanding initial conditions. Specifically, a cubic centimeter of a monatomic ideal gas of atomic weight 30 at room temperature and atmospheric pressure has about microstates, i.e., lives in a Hilbert space of that dimension. Squeezing them into 1/64 that volume (in coordinate space) changes the “20.28” to about 20.24. Squeezing them by only 1/8 and insisting that they reoccupy such a region again at a later time also brings the “20.28” to 20.24 [18]. There’s plenty of room in phase space.

#### 3.2 Isolation

#### 3.3 Closed timelike curves

#### 3.4 Entropy calculation

_{1},y

_{1},..., x

_{N}, y

_{N}), its coarse grained description is (n

_{1},..., n

_{G}), where n

_{k}is the number of atoms (x

_{ℓ},y

_{ℓ}) in grain k. Following the definition in [15], the entropy is , with p

_{k}= n

_{k}/N, if all grains are of equal coordinate space volume. The “p

_{k}log p

_{k}” as usual arises from the logarithm of N!/(n

_{1}!...n

_{G}!) and represents the missing information associated with particle identity. The missing information associated with going from real numbers to finite volumes is the same for all (n

_{1},..., n

_{G}), and is dropped, since in this study I am not concerned with comparing coarse grainings.

## 4. Causality

_{0}(0 < t

_{0}< T). So the m

**i**croscopic dynamics will in general differ at all times. What I looked at was the m

**a**croscopic behavior. And indeed, I found that if t

_{0}was close to 0, all macroscopic changes were confined to t > t

_{0}, while if t

_{0}is close to T, macroscopic changes were confined to t < t

_{0}, showing that “causality” follows the same arrow as entropy increase.

## Acknowledgments

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