Hydrodynamic fluctuations of entropy in one-dimensionally expanding system

The fluctuation-dissipation relation tells that dissipation always accompanies with thermal fluctuations. Relativistic fluctuating hydrodynamics is used to study the effects of the thermal fluctuations in the hydrodynamic expansion of the quark-gluon plasma created in the high-energy nuclear collisions. We show that the thermal noise obeys the steady-state fluctuation theorem when (i) the time scales of the evolution of thermodynamic quantities are sufficiently longer than the relaxation time, and (ii) the thermal fluctuations of temperature are sufficiently small. The steady-state fluctuation theorem describes the distribution of the entropy which can be related to the multiplicity observed in high-energy nuclear collisions. As a consequence, we propose an upper bound to the multiplicity fluctuations which is useful to test the initial state models. We also numerically investigate breaking of the steady-state fluctuation theorem due to the non-vanishing relaxation time in real nuclear collisions.