The absence of Bohr - van Leeuwen paradox in classical statistical ensambles of moving charges, in finite phase volume

TL;DR

This paper demonstrates that the Bohr-van Leeuwen theorem, which states classical systems cannot exhibit diamagnetism, only holds in infinite phase volume, and in finite phase volume, classical systems can have a diamagnetic orbital moment.

Contribution

It reveals that the Bohr-van Leeuwen paradox does not apply in finite phase volume, allowing classical diamagnetism to occur contrary to traditional understanding.

Findings

Bohr-van Leeuwen theorem applies only in infinite phase volume

Classical systems in finite phase volume can exhibit diamagnetism

The paradox is resolved by considering phase volume limitations

Abstract

The Bohr - van Leeuwen theorem [1-6] consists in follow paradox: classical statistical ensambles of moving charges in external static magnetic field can't have the induced orbital magnetic moment. I.e., the diamagnetism is not possible. In that paper will be shown, the theorem take place only in statistical ensambles in infinite phase volume. For the statistical ensambles in finite phase volume we have usual diamagnetic orbital moment.