Transition probability quantifies the likelihood of moving from one state to another in a stochastic process, providing critical insight into the dynamics of systems over time. It is a fundamental component in various fields such as Markov chains, quantum mechanics, and statistical mechanics, where it helps model and predict the behavior of complex systems.

Transition Probability

Transition probability

likelihood of moving from one state to another

A stochastic process is a collection of random variables representing the evolution of a system over time, where the future state depends on both the present state and inherent randomness. It is widely used in fields like finance, physics, and biology to model phenomena that evolve unpredictably over time.

stochastic process

dynamics of systems over time

fundamental component

A Markov Chain is a mathematical system that undergoes transitions from one state to another on a state space, where the probability of each state depends only on the state attained in the previous step. This 'memoryless' property, known as the Markov property, makes them particularly useful for modeling random processes in various fields such as economics, genetics, and computer science.

Markov chains

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the smallest scales, such as atoms and subatomic particles. It introduces concepts like wave-particle duality, uncertainty principle, and quantum entanglement, which challenge classical intuitions about the behavior of matter and energy.

quantum mechanics

Statistical mechanics is a branch of physics that uses probability theory to study and predict the behavior of systems with a large number of particles, bridging the microscopic laws of physics with macroscopic observations. It provides the framework for understanding thermodynamic properties and phase transitions by averaging over the possible states of a system rather than tracking individual particles.

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