In this work, we have rewritten the BDNK stress tensor in the Landau frame by
redefining the fluid variables such as velocity and temperature. This `fluid
frame’ transformation includes shift variables $delta u^{mu}$ and $delta T$,
which are small enough to be treated linearly but encompass all orders of
gradient corrections. The redefinition indicates that though the BDNK formalism
has a finite number of derivatives, in the Landau frame, it will have either an
infinite number of derivatives or one has to introduce new `non-fluid’
variables. The infinite derivative series are summed in two different ways that
lead to two different methods of `integrating in’ new `non-fluid’ variables,
showing the non-uniqueness of the process of `integrating in’ new variables.
Finally, the dispersion relations and the corresponding spectra of these
different systems of equations have been analyzed to check that the systems of
equations presented here are equivalent to the BDNK formalism, at least in the
hydrodynamic regime.

Future Roadmap: Challenges and Opportunities

Based on the conclusions of the study, several key findings and ideas for further exploration arise. This roadmap outlines potential challenges and opportunities that readers can expect in the future.

1. Further Investigation of the Redefined Fluid Variables

The redefinition of the fluid variables such as velocity and temperature in the Landau frame opens up new avenues for research. One potential challenge is to conduct a comprehensive analysis of these redefined variables to determine their impact on the BDNK formalism. Researchers can explore the implications of these changes for various physical processes and phenomena.

2. Incorporation of Non-Fluid Variables

The study suggests that in the Landau frame, the BDNK formalism may require the introduction of new ‘non-fluid’ variables. An opportunity lies in understanding the nature and role of these variables. Researchers can investigate how these additional variables affect the behavior and dynamics of the system under study. This exploration may lead to insights into previously unexplored aspects of the BDNK formalism.

3. Summation of Infinite Derivative Series

One significant finding is that the infinite derivative series can be summed in two different ways. This observation presents a challenge in understanding the underlying mathematics and physics behind these summation methods. Researchers can explore the implications of each method and analyze how they affect the overall behavior and properties of the system being studied.

4. Analysis of Dispersion Relations and Spectra

Future research should focus on analyzing the dispersion relations and corresponding spectra of the different systems of equations presented in this study. This analysis will help determine whether these new systems are equivalent to the original BDNK formalism, at least in the hydrodynamic regime. Understanding the similarities and differences between these systems is crucial for validating their applicability and extending the formalism to new regimes.


In conclusion, this study has introduced redefined fluid variables in the Landau frame and highlighted the need for including new ‘non-fluid’ variables. The existence of multiple summation methods for infinite derivative series and the analysis of dispersion relations provide exciting opportunities for further research. Addressing the challenges and exploring these opportunities will lead to a deeper understanding of the BDNK formalism and its implications across various physical systems.

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