  We calculate the off-equilibrium hydrostatic pressure for the O(N) Phi^{4}
theory to the leading order in 1/N. The present paper, the first of a series,
concentrates on the calculation of pressure in the non-equilibrium but
translationally invariant medium. The Jaynes-Gibbs principle of maximal entropy
is used to introduce the relevant density matrix which is then directly
implemented into dynamical equations through generalised Kubo-Martin-Schwinger
(KMS) conditions. We show that in the large N limit use of Ward identities
enables the pressure to be expressed in terms of two point Green's functions.
These satisfy the Kadanoff-Baym equations which are exactly solvable, and we
explicitly calculate the pressure for three illustrative choices of \rho.
