\[ N_\text{beds}=N_\text{occupied}+N_\text{available}=C \]
There also exists a bed conservation principle where for …
\[ N_\text{beds}=N_\text{occupied}+N_\text{available}=C \]
There also exists a bed conservation principle where for every discharge, there exists an equal and opposite admission
\[ \frac{dN_\text{occupied}}{dt}=\text{Rate}_\text{Admissions}-\text{Rate}_\text{Discharges}=0 \]
However, in reality \(\text{Rate}_\text{Admissions}>\text{Rate}_\text{Discharges}\), and therefore \(N_\text{Occupied}\to N_\text{Total}+\varepsilon\) where \(\varepsilon\) represents patients in beds in the hallway and new adhoc ‘short-stay’ locations. The hospital often violates this very conservation law.
Second Law: Entropy of Information Always Increases
\[ I(n)=I_0\times e^{-\lambda n} \] Where:
- \(I(n)\)
- information remaining after \(n\) retellings
- \(I_0\)
- Original information from consultant
- \(\lambda\)
- Decay constant (\(\approx 0.4\) per person)
- \(n\)
- Number of people in the chain
For a typical consult, the chain of information is Consultant → Registrar → Resident → Intern. Here \(I(3)\approx 0.30 \times I_0\) so only 30% of the original message is kept in the consult made by the intern. The other 70% is lost to entropy.
A useful corollary explores the death of the week explores the heat death of the weekend handover. Over the weekend Day team → Night team → Weekend team → Monday team. This is why Monday ward rounds feel like archeological expeditions through fragmentary records of decisions made by people who are no longer available.
Third Law: Approaching Absolute Zero
As the number of tasks approaches zero, the ward entropy (disorder/uncertainty) of the system approaches a minimum constant value, but never reaches zero.
\[ S = k_B \ln(\Omega) \]
Where:
- \(S\)
- Ward entropy (in units of “unpredictable events per shift”)
- \(k_B\)
- 1.38 (empirical constant relating microstates to macroscopic chaos)
- \(\Omega\)
- Number of independent ways the ward could surprise you
Components of \(\Omega\):
- \(N_\text{patients}\)
- Number of patients on ward
- \(P(\text{deterioration})\)
- Probability each patient deteriorates (≈ 0.1)
- \(N_\text{admin}\)
- Number of administrative systems that could malfunction
- \(N_\text{discharge}\)
- Number of discharge plans that could be complicated
- \(N_\text{handover}\)
- Number of handover points where information could be lost
Where for a typical thirty bed ward
\begin{align*} \Omega &\approx N_\text{patients} \times P(\text{deterioration}) + N_\text{admin} + N_\text{discharge} + N_\text{handover}\\ &\approx (30\times 0.1) +10 +8 + 12 = 33 \\ \therefore S &=\ln (33) \approx 3.5 \end{align*}
Which means that even with zero tasks, you can expect ~3-4 unexpected events per shift. Essentially, the third law means you cannot achieve a perfectly ordered ward (\(S=0\)) in finite time:
\[ \lim_{N_\text{tasks}\to 0} S = S_\text{residual} >0 \]
Anything can happen including patients deteriorating, eMR crashing, social issues arising on discharge, information lost during handover or random mandatory training modules. Peace in a hospital ward is thermodynamically forbidden.
Newton Laws of Ward Rounds
Fist Law: Inertia
A ward round in motion stays in motion until acted upon by an external force (usually coffee or bladders).
\[ v_\text{round} = \text{constant (unless interrupted)} \]
However, many stopping forces exist on a ward round including family at the bedside asking questions, consults coming in, consultants calls and other miscellaneous interruptions. Indeed these act as friction and the observable phenomena is that:
\[ v_\text{actual}=v_\text{planned}\times \prod_{i=0}^{N}(1-\text{interruption}_i) \]
Empirically, the value of the product of interruptions changes. It may be \(v_\text{actual}=0.4\times v_\text{planned}\) and so a ‘one-hour’ ward round becomes a two and a half hour ward round. This is not poor planning but just the physics of a ward round.
Second Law: Acceleration and Mass
The acceleration of the ward round is inversely proportional to the number of participants.
\[ a = \frac{F}{m} \]
Where:
- \(F\)
- Driving force (consultant urgency)
- \(m\)
- Mass of entourage \(\sum (\text{medical students} + \text{interns} + \text{residents} + \ldots)\)
In essence, the more people, the slower the round. This is why consultant-only rounds are fastest.
Third Law: Conservation of Responsibility
The responsibility of patient care cannot be destroyed, only redistributed. When a referral is made to another team, responsibility is now shared between the teams. When another team takes over care, the responsibility distribution shifts in favour of the new team but it is never destroyed. Even when a patient is discharged, the responsibility shifts to the care of the GP, the family and the community services.
Zeno’s Paradox of Discharge
Zeno of Elea famously proposed the Achilles and the tortoise. It is only fitting to name this discharge paradox after him.
The time required to discharge a patient increases exponentially with admission duration, approaching infinity as length of stay increases.
\[ T_\text{discharge} = T_\text{base} \times e^{\alpha \times \text{Length of Stay}} \times (1 + S + E + F) \]
Where:
- \(T_\text{discharge}\)
- Time until actual discharge
- \(T_\text{base}\)
- Baseline discharge time (≈ 2 days for simple medical patient)
- \(\alpha\)
- Complexity accumulation rate (≈ 0.15 per day)
- \(S\)
- Social issues factor (0 to ∞, typically 2-5)
- \(E\)
- Equipment delays (0-3)
- \(F\)
- Family meeting complexity (0-4)
For example, simple pneumonia, day 3: \[T = 2 \times e^{0.15 \times 3} \times (1 + 0 + 0 + 0) = 3.1 \text{ days}\]
Same patient, day 14, developed delirium, needs OT, family interstate: \[T = 2 \times e^{0.15 \times 14} \times (1 + 3 + 1.5 + 2.5) = 124 \text{ days}\]
The patient has crossed the event horizon. Discharge is now asymptotic.
The paradox remains in how to discharge, you must complete some amount the remaining tasks. Then some amount of what remains. All the while, the number of tasks to complete for discharge continues to increase. You approach discharge asymptotically but never quite arrive.
Conclusion
The hospital is a complex physical system governed by thermodynamic laws, force dynamics, and conservation principles. These laws are as fundamental as gravity—you can’t negotiate with them, you can only acknowledge them and adapt.
The physics of ward rounds is the physics of controlled chaos—where the laws of thermodynamics meet the realities of human systems, and somehow, despite everything, patients get better and go home.
Usually.
Eventually.
Asymptotically.
