We consider the fluctuations of a time-integrated particle current around an atypical value in a generic stochastic Markov process involving classical particles with two-site interaction and hardcore repulsion on a finite one-dimensional lattice with open boundaries. We address the question of which interactions one has to impose on such process to make the atypical value of the current typical. It is known that a corresponding effective stochastic Markov process might exist whose typical value of the current is equal to the atypical value of the current in the original process within a time-translational invariant regime. This effective process has, in principle, non-local transition rates. Nevertheless, it turns out that under some conditions the stochastic generator of the effective process has the same dynamical rules as the stochastic generator of the original process. We find these conditions and show that our approach can be generalized to any time-integrated observable.

We study the total particle current fluctuations in a one-dimensional stochastic system of classical particles consisting of branching and death processes which is a variant of asymmetric zerotemperature Glauber dynamics. The full spectrum of a modified Hamiltonian, whose minimum eigenvalue generates the large deviation function for the total particle current fluctuations through a Legendre-Fenchel transformation, is obtained analytically. Three examples are presented and numerically exact results are compared to our analytical calculations.

The dynamics of a one-dimensional stochastic system of classical particles consisting of asymmetric death and branching processes is studied. The dynamical activity, defined as the number of configuration changes in a dynamical trajectory, is considered as a proper dynamical order parameter. By considering an ensemble of dynamical trajectories and applying the large deviation method, we have found that the system might undergo both continuous and discontinuous dynamical phase transitions at critical values of the counting field. Exact analytical results are obtained for an infinite system. Numerical investigations confirm our analytical calculations.

We have considered a one-dimensional coagulation-decoagulation system of classical particles on a finite lattice with reflecting boundaries. It is known that the system undergoes a phase transition from a high-density to a low-density phase. Using a matrix product approach we have obtained an exact expression for the average entropy production rate of the system in the thermodynamic limit. We have also performed a large-deviation analysis for fluctuations of entropy production rate and particle current. It turns out that the characteristics of the kink in the large deviation function can be used to spot the phase transition point. We have found that for very weak driving field (when the system approaches its equilibrium) and also for very strong driving field (when the system is in the low-density phase) the large deviation function for fluctuations of entropy production rate is almost parabolic, while in the high-density phase it prominently deviates from Gaussian behavior. The validity of the Gallavotti-Cohen fluctuation relation for the large deviation function for particle current is also verified.

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