Non-monotonic Probability Research Articles

Non-monotonic probability of thermal reversal in thin-film biaxial nanomagnets with small energy barriers

The goal of this paper is to investigate the short time-scale, thermally-induced probability of magnetization reversal for an biaxial nanomagnet that is characterized with a biaxial magnetic anisotropy. For the first time, we clearly show that for a given energy barrier of the nanomagnet, the magnetization reversal probability of an biaxial nanomagnet exhibits a non-monotonic dependence on its saturation magnetization. Specifically, there are two reasons for this non-monotonic behavior in rectangular thin-film nanomagnets that have a large perpendicular magnetic anisotropy. First, a large perpendicular anisotropy lowers the precessional period of the magnetization making it more likely to precess across the x^=0 plane if the magnetization energy exceeds the energy barrier. Second, the thermal-field torque at a particular energy increases as the magnitude of the perpendicular anisotropy increases during the magnetization precession. This non-monotonic behavior is most noticeable when analyzing the magnetization reversals on time-scales up to several tens of ns. In light of the several proposals of spintronic devices that require data retention on time-scales up to 10’s of ns, understanding the probability of magnetization reversal on the short time-scales is important. As such, the results presented in this paper will be helpful in quantifying the reliability and noise sensitivity of spintronic devices in which thermal noise is inevitably present.

Non-monotonic probability theory for n-state quantum systems

In previous work, a non-standard theory of probability was formulated and used to systematize interference effects involving the simplest type of quantum systems. The main result here is a self-contained, non-trivial generalization of that theory to capture interference effects involving a much broader range of quantum systems. The discussion also focuses on interpretive matters having to do with the actual/virtual distinction, non-locality, and conditional probabilities.

Non-monotonic Probability Theory and Photon Polarization

A non-monotonic theory of probability is put forward and shown to have applicability in the quantum domain. It is obtained simply by replacing Kolmogorov’s positivity axiom, which places the lower bound for probabilities at zero, with an axiom that reduces that lower bound to minus one. Kolmogorov’s theory of probability is monotonic, meaning that the probability of A is less then or equal to that of B whenever A entails B. The new theory violates monotonicity, as its name suggests; yet, many standard theorems are also theorems of the new theory since Kolmogorov’s other axioms are retained. What is of particular interest is that the new theory can accommodate quantum phenomena (photon polarization experiments) while preserving Boolean operations, unlike Kolmogorov’s theory. Although non-standard notions of probability have been discussed extensively in the physics literature, they have received very little attention in the philosophical literature. One likely explanation for that difference is that their applicability is typically demonstrated in esoteric settings that involve technical complications. That barrier is effectively removed for non-monotonic probability theory by providing it with a homely setting in the quantum domain. Although the initial steps taken in this paper are quite substantial, there is much else to be done, such as demonstrating the applicability of non-monotonic probability theory to other quantum systems and elaborating the interpretive framework that is provisionally put forward here. Such matters will be developed in other works.

Adams Conditionals and Non-Monotonic Probabilities

Adams' famous thesis that the probabilities of conditionals are conditional probabilities is incompatible with standard probability theory. Indeed it is incompatible with any system of monotonic conditional probability satisfying the usual multiplication rule for conditional probabilities. This paper explores the possibility of accommodating Adams' thesis in systems of non-monotonic probability of varying strength. It shows that such systems impose many familiar lattice theoretic properties on their models as well as yielding interesting logics of conditionals, but that a standard complementation operation cannot be defined within them, on pain of collapsing probability into bivalence.